Number Theory for Mathematical Contests
David A. SANTOS
dsantos@ccp.edu
August 13, 2005
REVISION
Contents
iii 5
Preface
1 Preliminaries
1.1 Introduction . . . . . .
1.2 Well-Ordering . . . . .
Practice . . . . . . . . . . .
1.3 Mathematical Induction
Practice . . . . . . . . . . .
1.4 Fibonacci Numbers . .
Practice . . . . . . . . . . .
1.5 Pigeonhole Principle .
Practice . . . . . . . . . . .
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2 Divisibility
2.1 Divisibility . . . . . . . .
Practice . . . . . . . . . . . . .
2.2 Division Algorithm . . . .
Practice . . . . . . . . . . . . .
2.3 Some Algebraic Identities .
Practice . . . . . . . . . . . . .
3 Congruences. Zn
3.1 Congruences . . .
Practice . . . . . . . . .
3.2 Divisibility Tests .
Practice . . . . . . . . .
3.3 Complete Residues
Practice . . . . . . . . .
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4 Unique Factorisation
4.1 GCD and LCM . . . . . . . . . . .
Practice . . . . . . . . . . . . . . . . . .
4.2 Primes . . . . . . . . . . . . . . . .
Practice . . . . . . . . . . . . . . . . . .
4.3 Fundamental Theorem of Arithmetic
Practice . . . . . . . . . . . . . . . . . .
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1
1
1
3
3
7
9
11 6
13
14
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17
17
18
19
20
21
23
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26
26
30 7
31
32
33
33
8
34
34
38
39
41
41 9
45
Linear Diophantine Equations
5.1 Euclidean Algorithm . . . .
Practice . . . . . . . . . . . . . .
5.2 Linear Congruences . . . . .
Practice . . . . . . . . . . . . . .
5.3 A theorem of Frobenius . . .
Practice . . . . . . . . . . . . . .
5.4 Chinese Remainder Theorem
Practice . . . . . . . . . . . . . .
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48
48
50
51
52
52
54
55
56
Number-Theoretic Functions
6.1 Greatest Integer Function . . . . . .
Practice . . . . . . . . . . . . . . . . . .
6.2 De Polignac’s Formula . . . . . . .
Practice . . . . . . . . . . . . . . . . . .
6.3 Complementary Sequences . . . . .
Practice . . . . . . . . . . . . . . . . . .
6.4 Arithmetic Functions . . . . . . . .
Practice . . . . . . . . . . . . . . . . . .
6.5 Euler’s Function. Reduced Residues
Practice . . . . . . . . . . . . . . . . . .
6.6 Multiplication in Zn . . . . . . . . .
Practice . . . . . . . . . . . . . . . . . .
6.7 Möbius Function . . . . . . . . . .
Practice . . . . . . . . . . . . . . . . . .
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57
57
60
62
64
64
65
66
68
69
72
73
75
75
76
More on Congruences
7.1 Theorems of Fermat and Wilson
Practice . . . . . . . . . . . . . . . .
7.2 Euler’s Theorem . . . . . . . . .
Practice . . . . . . . . . . . . . . . .
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78
78
80
81
83
Scales of Notation
8.1 The Decimal Scale . .
Practice . . . . . . . . . . .
8.2 Non-decimal Scales . .
Practice . . . . . . . . . . .
8.3 A theorem of Kummer
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84
84
86
87
88
89
Miscellaneous Problems
Practice . . . . . . . . . . . . . . . . . . . . . . .
91
93
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Preface
These notes started in the summer of 1993 when I was teaching Number Theory at the Center for Talented Youth Summer
Program at the Johns Hopkins University. The pupils were between 13 and 16 years of age.
The purpose of the course was to familiarise the pupils with contest-type problem solving. Thus the majority of the problems are taken from well-known competitions:
AHSME
AIME
USAMO
IMO
ITT
MMPC
(UM)2
S TANFORD
M ANDELBROT
American High School Mathematics Examination
American Invitational Mathematics Examination
United States Mathematical Olympiad
International Mathematical Olympiad
International Tournament of Towns
Michigan Mathematics Prize Competition
University of Michigan Mathematics Competition
Stanford Mathematics Competition
Mandelbrot Competition
Firstly, I would like to thank the pioneers in that course: Samuel Chong, Nikhil Garg, Matthew Harris, Ryan Hoegg, Masha
Sapper, Andrew Trister, Nathaniel Wise and Andrew Wong. I would also like to thank the victims of the summer 1994: Karen
Acquista, Howard Bernstein, Geoffrey Cook, Hobart Lee, Nathan Lutchansky, David Ripley, Eduardo Rozo, and Victor Yang.
I would like to thank Eric Friedman for helping me with the typing, and Carlos Murillo for proofreading the notes.
Due to time constraints, these notes are rather sketchy. Most of the motivation was done in the classroom, in the notes
I presented a rather terse account of the solutions. I hope some day to be able to give more coherence to these notes. No
theme requires the knowledge of Calculus here, but some of the solutions given use it here and there. The reader not knowing
Calculus can skip these problems. Since the material is geared to High School students (talented ones, though) I assume very
little mathematical knowledge beyond Algebra and Trigonometry. Here and there some of the problems might use certain
properties of the complex numbers.
A note on the topic selection. I tried to cover most Number Theory that is useful in contests. I also wrote notes (which I
have not transcribed) dealing with primitive roots, quadratic reciprocity, diophantine equations, and the geometry of numbers.
I shall finish writing them when laziness leaves my weary soul.
I would be very glad to hear any comments, and please forward me any corrections or remarks on the material herein.
David A. SANTOS
dsantos@ccp.edu
iii
Legal Notice
This material may be distributed only subject to the terms and conditions set forth in the Open Publication License, version 1.0
or later (the latest version is presently available at
http://www.opencontent.org/openpub/ .
THIS WORK IS LICENSED AND PROVIDED “AS IS” WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
FOR A PARTICULAR PURPOSE OR A WARRANTY OF NON-INFRINGEMENT.
THIS DOCUMENT MAY NOT BE SOLD FOR PROFIT OR INCORPORATED INTO COMMERCIAL DOCUMENTS
WITHOUT EXPRESS PERMISSION FROM THE AUTHOR(S). THIS DOCUMENT MAY BE FREELY DISTRIBUTED
PROVIDED THE NAME OF THE ORIGINAL AUTHOR(S) IS(ARE) KEPT AND ANY CHANGES TO IT NOTED.
iv
Chapter
1
Preliminaries
1.1 Introduction
We can say that no history of mankind would ever be complete without a history of Mathematics. For ages numbers have
fascinated Man, who has been drawn to them either for their utility at solving practical problems (like those of measuring,
counting sheep, etc.) or as a fountain of solace.
Number Theory is one of the oldest and most beautiful branches of Mathematics. It abounds in problems that yet simple to
state, are very hard to solve. Some number-theoretic problems that are yet unsolved are:
1. (Goldbach’s Conjecture) Is every even integer greater than 2 the sum of distinct primes?
2. (Twin Prime Problem) Are there infinitely many primes p such that p + 2 is also a prime?
3. Are there infinitely many primes that are 1 more than the square of an integer?
4. Is there always a prime between two consecutive squares of integers?
In this chapter we cover some preliminary tools we need before embarking into the core of Number Theory.
1.2 Well-Ordering
The set N = {0, 1, 2, 3, 4, . . .} of natural numbers is endowed with two operations, addition and multiplication, that satisfy the
following properties for natural numbers a, b, and c:
1. Closure: a + b and ab are also natural numbers.
2. Associative laws: (a + b) + c = a + (b + c) and a(bc) = (ab)c.
3. Distributive law: a(b + c) = ab + ac.
4. Additive Identity: 0 + a = a + 0 = a
5. Multiplicative Identity: 1a = a1 = a.
One further property of the natural numbers is the following.
1 Axiom (Well-Ordering Axiom) Every non-empty subset S of the natural numbers has a least element.
As an example of the use of the Well-Ordering Axiom, let us prove that there is no integer between 0 and 1.
2 Example Prove that there is no integer in the interval ]0; 1[.
1
2
Chapter 1
Solution: Assume to the contrary that the set S of integers in ]0; 1[ is non-empty. Being a set of positive integers, it must
contain a least element, say m. Now, 0 < m2 < m < 1, and so m2 ∈ S . But this is saying that S has a positive integer m2
which is smaller than its least positive integer m. This is a contradiction and so S = ∅.
We denote the set of all integers by Z, i.e.,
Z = {. . . − 3, −2, −1, 0, 1, 2, 3, . . .}.
a
A rational number is a number which can be expressed as the ratio of two integers a, b, where b 6= 0. We denote the set of
b
rational numbers by Q. An irrational number is a number which cannot be expressed as the ratio of two integers. Let us give
an example of an irrational number.
3 Example Prove that
√
2 is irrational.
√
√
a
Solution: The proof is by contradiction. Suppose that 2 were rational, i.e., that 2 = for some integers a, b. This implies
b
that the set
√
√
A = {n 2 : both n and n 2 positive integers}
√
√
is nonempty since it contains a. By Well-Ordering A has a smallest element, say j = k 2. As 2 − 1 > 0,
√
√
√
√
j( 2 − 1) = j 2 − k 2 = ( j − k) 2
√
√
√
√
is a positive integer. Since 2 < 2 2 implies 2 − 2 < 2 and also j 2 = 2k, we see that
√
√
√
( j − k) 2 = k(2 − 2) < k( 2) = j.
√
Thus ( j − k) 2 is a positive integer in A which is smaller than j. This contradicts the choice of j as the smallest integer in A
and hence, finishes the proof.
4 Example Let a, b, c be integers such that a6 + 2b6 = 4c6 . Show that a = b = c = 0.
Solution: Clearly we can restrict ourselves to nonnegative numbers. Choose a triplet of nonnegative integers a, b, c satisfying
this equation and with
max(a, b, c) > 0
as small as possible. If a6 + 2b6 = 4c6 then a must be even, a = 2a1 . This leads to 32a61 + b6 = 2c6 . Hence b = 2b1 and so
16a61 + 32b61 = c6 . This gives c = 2c1 , and so a61 + 2b61 = 4c61 . But clearly max(a1 , b1 , c1 ) < max(a, b, c). This means that all of
these must be zero.
5 Example (IMO 1988) If a, b are positive integers such that
a2 + b2
a2 + b2
is an integer, then
is a perfect square.
1 + ab
1 + ab
a2 + b2
= k is a counterexample of an integer which is not a perfect square, with max(a, b) as small as
1 + ab
possible. We may assume without loss of generality that a < b for if a = b then
Solution: Suppose that
0<k=
2a2
< 2,
a2 + 1
which forces k = 1, a perfect square.
Now, a2 + b2 − k(ab + 1) = 0 is a quadratic in b with sum of the roots ka and product of the roots a2 − k. Let b1 , b be its
roots, so b1 + b = ka and b1 b = a2 − k.
As a, k are positive integers, supposing b1 < 0 is incompatible with a2 + b21 = k(ab1 + 1). As k is not a perfect square,
supposing b1 = 0 is incompatible with a2 + 02 = k(0 · a + 1). Also
b1 =
a2 − k b2 − k
<
< b.
b
b
Practice
3
a2 + b21
= k and which is smaller than the smallest max(a, b). This
1 + ab1
is a contradiction. It must be the case, then, that k is a perfect square.
Thus we have found another positive integer b1 for which
Practice
6 Problem Find all integer solutions of a3 + 2b3 = 4c3 .
7 Problem Prove that the equality x2 +y2 +z2 = 2xyz can hold
for whole numbers x, y, z only when x = y = z = 0.
1.3 Mathematical Induction
The Principle of Mathematical Induction is based on the following fairly intuitive observation. Suppose that we are to perform
a task that involves a certain number of steps. Suppose that these steps must be followed in strict numerical order. Finally,
suppose that we know how to perform the n-th task provided we have accomplished the n − 1-th task. Thus if we are ever able
to start the job (that is, if we have a base case), then we should be able to finish it (because starting with the base case we go to
the next case, and then to the case following that, etc.).
Thus in the Principle of Mathematical Induction, we try to verify that some assertion P(n) concerning natural numbers is
true for some base case k0 (usually k0 = 1, but one of the examples below shows that we may take, say k0 = 33.) Then we try
to settle whether information on P(n − 1) leads to favourable information on P(n).
We will now derive the Principle of Mathematical Induction from the Well-Ordering Axiom.
8 Theorem (Principle of Mathematical Induction) If a setS of non-negative integers contains the integer 0, and also contains the integer n + 1 whenever it contains the integer n, then S = N.
Proof: Assume this is not the case and so, by the Well-Ordering Principle there exists a least positive integer k
not in S . Observe that k > 0, since 0 ∈ S and there is no positive integer smaller than 0. As k − 1 < k, we see that
k − 1 ∈ S . But by assumption k − 1 + 1 is also in S , since the successor of each element in the set is also in the
set. Hence k = k − 1 + 1 is also in the set, a contradiction. Thus S = N. ❑
The following versions of the Principle of Mathematical Induction should now be obvious.
9 Corollary If a set A of positive integers contains the integer m and also contains n + 1 whenever it contains n, where n > m,
then A contains all the positive integers greater than or equal to m.
10 Corollary (Principle of Strong Mathematical Induction) If a set A of positive integers contains the integer m and also
contains n + 1 whenever it contains m + 1, m + 2, . . . , n, where n > m, then A contains all the positive integers greater than or
equal to m.
We shall now give some examples of the use of induction.
11 Example Prove that the expression
33n+3 − 26n − 27
is a multiple of 169 for all natural numbers n.
Solution: For n = 1 we are asserting that 36 − 53 = 676 = 169 · 4 is divisible by 169, which is evident. Assume the assertion is
true for n − 1, n > 1, i.e., assume that
33n − 26n − 1 = 169N
for some integer N. Then
33n+3 − 26n − 27 = 27 · 33n − 26n − 27 = 27(33n − 26n − 1) + 676n
4
Chapter 1
which reduces to
27 · 169N + 169 · 4n,
which is divisible by 169. The assertion is thus established by induction.
12 Example Prove that
√
√
(1 + 2)2n + (1 − 2)2n
is an even integer and that
√
√
√
(1 + 2)2n − (1 − 2)2n = b 2
for some positive integer b, for all integers n ≥ 1.
√ 2n
√ 2n
√ 2n
Solution:
We
√ proceed by induction on n. Let P(n) be the proposition: “(1 + 2) + (1 − 2) is even and (1 + 2) − (1 −
√ 2n
2) = b 2 for some b ∈ N.” If n = 1, then we see that
√
√
(1 + 2)2 + (1 − 2)2 = 6,
an even integer, and
√
√
√
(1 + 2)2 − (1 − 2)2 = 4 2.
Therefore P(1) is true. Assume that P(n − 1) is true for n > 1, i.e., assume that
√
√
(1 + 2)2(n−1) + (1 − 2)2(n−1) = 2N
for some integer N and that
√
√
√
(1 + 2)2(n−1) − (1 − 2)2(n−1) = a 2
for some positive integer a.
Consider now the quantity
√
√
√
√
√
√
(1 + 2)2n + (1 − 2)2n = (1 + 2)2 (1 + 2)2n−2 + (1 − 2)2 (1 − 2)2n−2 .
This simplifies to
√
√
√
√
(3 + 2 2)(1 + 2)2n−2 + (3 − 2 2)(1 − 2)2n−2 .
Using P(n − 1), the above simplifies to
√ √
12N + 2 2a 2 = 2(6N + 2a),
an even integer and similarly
√
√
√
√
√
(1 + 2)2n − (1 − 2)2n = 3a 2 + 2 2(2N) = (3a + 4N) 2,
and so P(n) is true. The assertion is thus established by induction.
13 Example Prove that if k is odd, then 2n+2 divides
n
k2 − 1
for all natural numbers n.
Solution: The statement is evident for n = 1, as k2 − 1 = (k − 1)(k + 1) is divisible by 8 for any odd natural number k because
n
both (k − 1) and (k + 1) are divisible by 2 and one of them is divisible by 4. Assume that 2n+2 |k2 − 1, and let us prove that
n+1
n+1
n
n
2n+3 |k2 − 1. As k2 − 1 = (k2 − 1)(k2 + 1), we see that 2n+2 divides (k2n − 1), so the problem reduces to proving that
2n
2n
2|(k + 1). This is obviously true since k odd makes k2n + 1 even.
Mathematical Induction
5
14 Example (USAMO 1978) An integer n will be called good if we can write
n = a1 + a2 + · · · + ak ,
where a1 , a2 , . . . , ak are positive integers (not necessarily distinct) satisfying
1
1
1
+ + · · · + = 1.
a1 a2
ak
Given the information that the integers 33 through 73 are good, prove that every integer ≥ 33 is good.
Solution: We first prove that if n is good, then 2n + 8 and 2n + 9 are good. For assume that n = a1 + a2 + · · · + ak , and
1=
1
1
1
+ +···+ .
a1 a2
ak
Then 2n + 8 = 2a1 + 2a2 + · · · + 2ak + 4 + 4 and
1
1
1
1 1 1 1 1
+
+···+
+ + = + + = 1.
2a1 2a2
2ak 4 4 2 4 4
Also, 2n + 9 = 2a1 + 2a2 + · · · + 2ak + 3 + 6 and
1
1
1 1 1 1 1
1
+
+···+
+ + = + + = 1.
2a1 2a2
2ak 3 6 2 3 6
Therefore,
if n is good both 2n + 8 and 2n + 9 are good.
(1.1)
We now establish the truth of the assertion of the problem by induction on n. Let P(n) be the proposition “all the integers
n, n + 1, n + 2, . . . , 2n + 7” are good. By the statement of the problem, we see that P(33) is true. But (1.1) implies the truth of
P(n + 1) whenever P(n) is true. The assertion is thus proved by induction.
We now present a variant of the Principle of Mathematical Induction used by Cauchy to prove the Arithmetic-MeanGeometric Mean Inequality. It consists in proving a statement first for powers of 2 and then interpolating between powers of
2.
15 Theorem (Arithmetic-Mean-Geometric-Mean Inequality) Let a1 , a2 , . . . , an be nonnegative real numbers. Then
√
a1 + a2 + · · · + an
n
.
a1 a2 · · · an ≤
n
Proof: Since the square of any real number is nonnegative, we have
√
√
( x1 − x2 )2 ≥ 0.
Upon expanding,
x1 + x2 √
≥ x1 x2 ,
(1.2)
2
which is the Arithmetic-Mean-Geometric-Mean Inequality for n = 2. Assume that the Arithmetic-Mean-GeometricMean Inequality holds true for n = 2k−1 , k > 2, that is, assume that nonnegative real numbers w1 , w2 , . . . , w2k−1
satisfy
k−1
w1 + w2 + · · · + w2k−1
≥ (w1 w2 · · · w2k−1 )1/2 .
(1.3)
k−1
2
Using (1.2) with
y1 + y2 + · · · + y2k−1
x1 =
2k−1
and
y k−1 + · · · + y2k
x2 = 2 +1 k−1
,
2
6
Chapter 1
we obtain that
y1 + y2 + · · · + y2k−1 y2k−1 +1 + · · · + y2k
+
2k−1
2k−1
≥
2

‹1/2
(
y1 + y2 + · · · + y2k−1 y2k−1 +1 + · · · + y2k
)(
)
2k−1
2k−1
.
Applying (1.3) to both factors on the right hand side of the above , we obtain
k
y1 + y2 + · · · + y2k
≥ (y1 y2 · · · y2k )1/2 .
2k
(1.4)
This means that the 2k−1 -th step implies the 2k -th step, and so we have proved the Arithmetic-Mean-GeometricMean Inequality for powers of 2.
Now, assume that 2k−1 < n < 2k . Let
y1 = a1 , y2 = a2 , . . . , yn = an ,
and
yn+1 = yn+2 = · · · = y2k =
Let
A=
a1 + a2 + · · · + an
.
n
a1 + · · · + an
and G = (a1 · · · an )1/n .
n
Using (1.4) we obtain
a1 + a2 + · · · + an + (2k − n)
2k
a1 + · · · + an
n
≥
a1 + · · · + an (2k −n) 1/2
,
)
n
k
a1 a2 · · · an (
which is to say that
k
k
nA + (2k − n)A
≥ (Gn A2 −n )1/2 .
2k
This translates into A ≥ G or
(a1 a2 · · · an )1/n ≤
a1 + a2 + · · · + an
,
n
which is what we wanted.❑
16 Example Let s be a positive integer. Prove that every interval [s; 2s] contains a power of 2.
Solution: If s is a power of 2, then there is nothing to prove. If s is not a power of 2 then it must lie between two consecutive
powers of 2, i.e., there is an integer r for which 2r < s < 2r+1 . This yields 2r+1 < 2s. Hence s < 2r+1 < 2s, which gives the
required result.
√
17 Example Let M be a nonempty set of positive integers such that 4x and [ x] both belong to M whenever x does. Prove
that M is the set of all natural numbers.
Solution: We will prove this by induction. First we will prove that 1 belongs to the set, secondly we will prove that every power
of 2 is in the set and finally we will prove that non-powers of 2 are also in the set.
√
√ Since M is a nonempty set of positive integers, it has a least element, say a. By assumption T aU also belongs to M , but
a < a unless a = 1. This means that 1 belongs to M .
Since 1 belongs to M so does 4, since 4 belongs to M so does 4 · 4 = 42 , etc.. In this way we obtain that all numbers of
the form 4n = 22n , n = 1, 2, . . . belong to M . Thus all the powers of 2 raised to an even power belong to M . Since the square
roots belong as well to M we get that all the powers of 2 raised to an odd power also belong to M . In conclusion, all powers
of 2 belong to M .
Practice
7
Assume now that n ∈ N fails to belong to M . Observe that n cannot be a power of 2. Since n 6∈ M we deduce that
√
no integer in A1 = [n2 , (n + 1)2 ) belongs to M , because every member of y ∈ A1 satisfies [ y] = n. Similarly no member
z ∈ A2 = [n4 , (n + 1)4 ) belongs to M since this would entail that z would belong to A1 , a contradiction. By induction we can
r
r
show that no member in the interval Ar = [n2 , (n + 1)2 ) belongs to M .
We will now show that eventually these intervals are so large that they contain a power of 2, thereby obtaining a contradiction
to the hypothesis that no element of the Ar belonged to M . The function
R∗+
x
f:
→
7
→
R
log2 x
is increasing and hence log2 (n + 1) − log2 n > 0. Since the function
f:
R →
x 7→
R∗+
2−x
is decreasing, for a sufficiently large positive integer k we have
2−k < log2 (n + 1) − log2 n.
This implies that
k
k
(n + 1)2 > 2n2 .
k
k
k
k
Thus the interval [n2 , 2n2 ] is totally contained in [n2 , (n + 1)2 ). But every interval of the form [s, 2s] where s is a positive
integer contains a power of 2. We have thus obtained the desired contradiction.
Practice
18 Problem Prove that 11n+2 + 122n+1 is divisible by 133 for
22 Problem Let a1 = 3, b1 = 4, and an = 3an−1 , bn = 4bn−1
all natural numbers n.
when n > 1. Prove that a1000 > b999 .
19 Problem Prove that
23 Problem Let n ∈ N, n > 1. Prove that
1−
x x(x − 1) x(x − 1)(x − 2)
+
−
1!
2!
3!
+ · · · + (−1)n
x(x − 1)(x − 2) · · · (x − n + 1)
n!
equals
n (x − 1)(x − 2) · · · (x − n)
(−1)
20 Problem Let n ∈ N. Prove the inequality
1
1
1
+
+···+
> 1.
n+1 n+2
3n + 1
21 Problem Prove that
|
È
2+
√
π
2 + · · · + 2 = 2 cos n+1
{z
}
2
n radical signs
for n ∈ N.
24 Problem Prove that if n is a natural number, then
1 · 2 + 2 · 5 + · · · + n · (3n − 1) = n2 (n + 1).
n!
for all non-negative integers n.
q
1 · 3 · 5 · · · (2n − 1)
1
.
<√
2 · 4 · 6 · · · (2n)
3n + 1
25 Problem Prove that if n is a natural number, then
12 + 32 + 52 + · · · + (2n − 1)2 =
n(4n2 − 1)
.
3
26 Problem Prove that
4n
(2n)!
<
n+1
(n!)2
for all natural numbers n > 1.
27 Problem Prove that the sum of the cubes of three consecutive positive integers is divisible by 9.
8
Chapter 1
28 Problem If |x| =
6 1, n ∈ N prove that
3. Prove that if n > 1, then
1 · 3 · 5 · · · (2n − 1) < nn .
1
2
4
8
2n
+
+
+
+
·
·
·
+
n
1 + x 1 + x2 1 + x2 1 + x8
1 + x2
4. Prove that if n > 1 then
equals
€
1
2n+1
+
.
x − 1 1 − x2n+1
Š
n (n + 1)1/n − 1 < 1 +
1
1
+···+ .
2
n
5. Prove that if n > 1 then
29 Problem Is it true that for every natural number n the
quantity n2 + n + 41 is a prime? Prove or disprove!
30 Problem Give an example of an assertion which is not true
for any positive integer, yet for which the induction step holds.
1+
6. Given that u, v, w are positive, 0 < a ≤ 1, and that
u + v + w = 1, prove that

31 Problem Give an example of an assertion which is true for
the first two million positive integers but fails for every integer
greater than 2000000.
32 Problem Prove by induction on n that a set having n elements has exactly 2n subsets.
33 Problem Prove that if n is a natural number,
n5 /5 + n4 /2 + n3 /3 − n/30
is always an integer.
1
1
1
1
+···+ < n 1−
.
+
1/n
2
n
n+1
(n + 1)
‹
1
−a
u
‹
1
−a
v
‹
1
− a ≥ 27 − 27a + 9a2 − a3 .
w
7. Let y1 , y2 , . . . , yn be positive real numbers. Prove the
Harmonic-Mean- Geometric-Mean Inequality:
n
1
1
1
+ +···+
y1 y2
yn
≤
√
n y y ···y .
n
1 2
8. Let a1 , . . . , an be positive real numbers, all different. Set
s = a1 + a2 + · · · + an .
(a) Prove that
34 Problem (Halmos) ) Every man in a village knows in-
(n − 1)
X
1
<
s − ar
X
1
.
ar
1≤r≤n
1≤r≤n
stantly when another’s wife is unfaithful, but never when his
own is. Each man is completely intelligent and knows that ev(b) Deduce that
ery other man is. The law of the village demands that when
X
n X 1
1
4n
a man can PROVE that his wife has been unfaithful, he must
<s
<
.
shoot her before sundown the same day. Every man is coms
a (s − ar ) n − 1
a
1≤r≤n r
1≤r≤n r
pletely law-abiding. One day the mayor announces that there
is at least one unfaithful wife in the village. The mayor always
tells the truth, and every man believes him. If in fact there 36 Problem Suppose that x1 , x2 , . . . , xn are nonnegative real
are exactly forty unfaithful wives in the village (but that fact numbers with
x1 + x2 + · · · + xn ≤ 1/2.
is not known to the men,) what will happen after the mayor’s
announcement?
Prove that
35 Problem
1. Let a1 , a2 , . . . an be positive real numbers
(1 − x1 )(1 − x2 ) · · · (1 − xn ) ≥ 1/2.
with
a1 · a2 · · · an = 1.
Use induction to prove that
a1 + a2 + · · · + an ≥ n,
with equality if and only if a1 = a2 = · · · = an = 1.
37 Problem Given a positive integer n prove that there is a
polynomial Tn such that cos nx = Tn (cos x) for all real numbers
x. Tn is called the n-th Tchebychev Polynomial.
38 Problem Prove that
1
1
1
13
+
+···+
>
n+1 n+2
2n 24
2. Use the preceding part to give another proof of the
for all natural numbers n > 1.
Arithmetic-Mean-Geometric-Mean Inequality.
Fibonacci Numbers
9
39 Problem In how many regions will a sphere be divided
by n planes passing through its centre if no three planes pass
through one and the same diameter?
41 Problem Let F0 (x) = x, F(x) = 4x(1 − x), Fn+1 (x) =
F(Fn (x)), n = 0, 1, . . . . Prove that
Z 1
Fn (x) dx =
40 Problem (IMO 1977) Let f , f : N 7→ N be a function satis-
0
fying
f (n + 1) > f ( f (n))
for each positive integer n. Prove that f (n) = n for each n.
22n−1
.
22n − 1
(Hint: Let x = sin2 θ .)
1.4 Fibonacci Numbers
The Fibonacci numbers fn are given by the recurrence
f0 = 0, f1 = 1, fn+1 = fn−1 + fn , n ≥ 1.
(1.5)
Thus the first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, . . . . A number of interesting algebraic identities can be
proved using the above recursion.
42 Example Prove that
f1 + f2 + · · · + fn = fn+2 − 1.
Solution: We have
f1
f2
f3
..
.
= f3 − f2
= f4 − f3
= f5 − f4
..
.
fn
= fn+2 − fn+1
Summing both columns,
f1 + f2 + · · · + fn = fn+2 − f2 = fn+2 − 1,
as desired.
43 Example Prove that
f1 + f3 + f5 + · · · + f2n−1 = f2n .
Solution: Observe that
f1
f3
f5
..
.
=
=
=
..
.
f2 − f0
f4 − f2
f6 − f4
..
.
f2n−1
=
f2n − f2n−2
Adding columnwise we obtain the desired identity.
44 Example Prove that
f12 + f22 + · · · + fn2 = fn fn+1 .
Solution: We have
fn−1 fn+1 = ( fn+1 − fn )( fn + fn−1 ) = fn+1 fn − fn2 + fn+1 fn−1 − fn fn−1 .
Thus
fn+1 fn − fn fn−1 = fn2 ,
10
Chapter 1
which yields
f12 + f22 + · · · + fn2 = fn fn+1 .
45 Theorem (Cassini’s Identity)
fn−1 fn+1 − fn2 = (−1)n , n ≥ 1.
Proof: Observe that
fn−1 fn+1 − fn2
= ( fn − fn−2 )( fn + fn−1 ) − fn2
= − fn−2 fn − fn−1 ( fn−2 − fn )
2
= −( fn−2 fn − fn−1
)
Thus if vn = fn−1 fn+1 − fn2 , we have vn = −vn−1 . This yields vn = (−1)n−1 v1 which is to say
fn−1 fn+1 − fn2 = (−1)n−1 ( f0 f2 − f12 ) = (−1)n .
❑
46 Example (IMO 1981) Determine the maximum value of
m2 + n2 ,
where m, n are positive integers satisfying m, n ∈ {1, 2, 3, . . . , 1981} and
(n2 − mn − m2 )2 = 1.
Solution: Call a pair (n, m) admissible if m, n ∈ {1, 2, . . . , 1981} and (n2 − mn − m2 )2 = 1.
If m = 1, then (1, 1) and (2, 1) are the only admissible pairs. Suppose now that the pair (n1 , n2 ) is admissible, with n2 > 1.
As n1 (n1 − n2 ) = n22 ± 1 > 0, we must have n1 > n2 .
Let now n3 = n1 − n2 . Then 1 = (n21 − n1 n2 − n22 )2 = (n22 − n2 n3 − n23 )2 , making (n2 , n3 ) also admissible. If n3 > 1, in the
same way we conclude that n2 > n3 and we can let n4 = n2 − n3 making (n3 , n4 ) an admissible pair. We have a sequence of
positive integers n1 > n2 > . . ., which must necessarily terminate. This terminates when nk = 1 for some k. Since (nk−1 , 1)
is admissible, we must have nk−1 = 2. The sequence goes thus 1, 2, 3, 5, 8, . . . , 987, 1597, i.e., a truncated Fibonacci sequence.
2
2
The largest admissible
pair is thus (1597, 987) and so the maximum
√
√ sought is 1597 + 987 .
1+ 5
5−1
be the Golden Ratio. Observe that τ −1 =
. The number τ is a root of the quadratic equation
Let τ =
2
2
2
x = x + 1. We now obtain a closed formula for fn . We need the following lemma.
47 Lemma If x2 = x + 1, n ≥ 2 then we have xn = fn x + fn−1 .
Proof: We prove this by induction on n. For n = 2 the assertion is a triviality. Assume that n > 2 and that
xn−1 = fn−1 x + fn−2 . Then
xn = xn−1 · x
= ( fn−1 x + fn−2 )x
= fn−1 (x + 1) + fn−2 x
= ( fn−1 + fn−2 )x + fn−1
= fn x + fn−1
❑
48 Theorem (Binet’s Formula) The n-th Fibonacci number is given by
1
fn = √
5
n = 0, 2, . . . .
‚‚
√ Œn ‚
√ Œn Œ
1+ 5
1− 5
−
2
2
Practice
11
√
√
1+ 5
1− 5
Proof: The roots of the equation x = x + 1 are τ =
and 1 − τ =
. In virtue of the above lemma,
2
2
2
τ n = τ fn + fn−1
and
(1 − τ )n = (1 − τ ) fn + fn−1 .
Subtracting
τ n − (1 − τ )n =
√
5 fn ,
from where Binet’s Formula follows.❑
49 Example (Cesàro) Prove that
n
X
k=0
n k
2 fk = f3n .
k
Solution: Using Binet’s Formula,
n
X
k=0
n k
2 fk
k
=
n
X
k=0
=
=
n k τ k − (1 − τ )k
√
2
k
5
n n 1 X n k X n k
√
τ −
2 (1 − τ )k
k
5 k=0 k
k=0
1
√ ((1 + 2τ )n − (1 + 2(1 − τ ))n ) .
5
!
As τ 2 = τ + 1, 1 + 2τ = τ 3 . Similarly 1 + 2(1 − τ ) = (1 − τ )3 . Thus
n
X
k=0
Š
n k
1 €
2 fk = √ (τ )3n + (1 − τ )3n = f3n ,
k
5
as wanted.
The following theorem will be used later.
50 Theorem If s ≥ 1,t ≥ 0 are integers then
fs+t = fs−1 ft + fs ft+1 .
Proof: We keep t fixed and prove this by using strong induction on s. For s = 1 we are asking whether
ft+1 = f0 ft + f1 ft+1 ,
which is trivially true. Assume that s > 1 and that fs−k+t = fs−k−1 ft + fs−k ft+1 for all k satisfying 1 ≤ k ≤ s − 1.
We have
fs+t = fs+t−1 + fs+t−2
by the Fibonacci recursion,
= fs−1+t + fs−2+t
trivially,
= fs−2 ft + fs−1 ft+1 + fs−3 ft + fs−2 ft+1 by the inductive assumption
= ft ( fs−2 + fs−3 ) + ft+1 ( fs−1 + fs−2 )
rearranging,
= ft fs−1 + ft+1 fs
by the Fibonacci recursion.
This finishes the proof.❑
Practice
12
Chapter 1
51 Problem Prove that
61 Problem Prove that
∞
X
fn+1 fn − fn−1 fn−2 = f2n−1 , n > 2.
arctan
n=1
52 Problem Prove that
2
2
fn+1
= 4 fn fn−1 + fn−2
, n > 1.
62 Problem Prove that
1
fn
=√ .
n
n→∞ τ
5
53 Problem Prove that
lim
2
f1 f2 + f2 f3 + · · · + f2n−1 f2n = f2n
.
54 Problem Let N be a natural number. Prove that the largest
n such that fn ≤ N is given by

63 Problem Prove that
lim
‹
1 √
log N +
5
2 Œ
‚
√
U.
n=T
1+ 5
log
2
55 Problem Prove that
1
= π /4.
f2n+1
2
fn2 + fn−1
=
fn+r
= τr.
fn
64 Problem Prove that
n
X
= f2n+1 .
f2n −2
1
= 2+
.
f 2k
f 2n
k=0
Deduce that
56 Problem Prove that if n > 1,
fn2 − fn+l fn−l
n→∞
∞
X
(−1)n+l fl2 .
k=0
√
7− 5
1
=
.
f 2k
2
57 Problem Prove that
n
X
f2k =
k=1
65 Problem (Cesàro) Prove that
n
X
(n − k) f2k+1 .
n
X
k=0
k=0
58 Problem Prove that
∞
X
n=2
Hint: What is
1
= 1.
fn−1 fn+1
1
fn−1 fn
−
1
?
fn fn+1
59 Problem Prove that
∞
X
n=1
n
fk = f2n .
k
66 Problem Prove that
∞
X
n=1
fn
10n
is a rational number.
67 Problem Find the exact value of
fn
= 1.
fn+1 fn+2
1994
X
(−1)k
k=1
1995
fk .
k
60 Problem Prove that
∞
X
n=0
1/ f2n = 4 − τ .
68 Problem Prove the converse of Cassini’s Identity: If k and
m are integers such that |m2 − km − k2 | = 1, then there is an
integer n such that k = ± fn , m = ± fn+1 .
Pigeonhole Principle
13
1.5 Pigeonhole Principle
The Pigeonhole Principle states that if n + 1 pigeons fly to n holes, there must be a pigeonhole containing at least two pigeons.
This apparently trivial principle is very powerful. Let us see some examples.
69 Example (Putnam 1978) Let A be any set of twenty integers chosen from the arithmetic progression 1, 4, . . . , 100. Prove
that there must be two distinct integers in A whose sum is 104.
Solution: We partition the thirty four elements of this progression into nineteen groups {1}, {52}, {4, 100} , {7, 97}, {10, 94},
. . . {49, 55}. Since we are choosing twenty integers and we have nineteen sets, by the Pigeonhole Principle there must be two
integers that belong to one of the pairs, which add to 104.
70 Example Show that amongst any seven distinct positive integers not exceeding 126, one can find two of them, say a and b,
which satisfy
b < a ≤ 2b.
Solution: Split the numbers {1, 2, 3, . . . , 126} into the six sets
{1, 2}, {3, 4, 5, 6}, {7, 8, . . . , 13, 14}, {15, 16, . . . , 29, 30},
{31, 32, . . . , 61, 62} and {63, 64, . . . , 126}.
By the Pigeonhole Principle, two of the seven numbers must lie in one of the six sets, and obviously, any such two will satisfy
the stated inequality.
71 Example Given any set of ten natural numbers between 1 and 99 inclusive, prove that there are two disjoint nonempty
subsets of the set with equal sums of their elements.
Solution: There are 210 − 1 = 1023 non-empty subsets that one can form with a given 10-element set. To each of these subsets
we associate the sum of its elements. The maximum value that any such sum can achieve is 90 + 91 + · · · + 99 = 945 < 1023.
Therefore, there must be at least two different subsets that have the same sum.
72 Example No matter which fifty five integers may be selected from
{1, 2, . . . , 100},
prove that one must select some two that differ by 10.
Solution: First observe that if we choose n + 1 integers from any string of 2n consecutive integers, there will always be some
two that differ by n. This is because we can pair the 2n consecutive integers
{a + 1, a + 2, a + 3, . . . , a + 2n}
into the n pairs
{a + 1, a + n + 1}, {a + 2, a + n + 2}, . . . , {a + n, a + 2n},
and if n + 1 integers are chosen from this, there must be two that belong to the same group.
So now group the one hundred integers as follows:
{1, 2, . . . 20}, {21, 22, . . . , 40},
{41, 42, . . . , 60}, {61, 62, . . . , 80}
and
{81, 82, . . . , 100}.
If we select fifty five integers, we must perforce choose eleven from some group. From that group, by the above observation
(let n = 10), there must be two that differ by 10.
14
Chapter 1
73 Example (AHSME 1994) Label one disc “1”, two discs “2”, three discs “3”, . . . , fifty discs ‘‘50”. Put these 1 + 2 + 3 + · · · +
50 = 1275 labeled discs in a box. Discs are then drawn from the box at random without replacement. What is the minimum
number of discs that must me drawn in order to guarantee drawing at least ten discs with the same label?
Solution: If we draw all the 1 + 2 + · · · + 9 = 45 labelled “1”, . . . , “9” and any nine from each of the discs “10”, . . . , “50”, we
have drawn 45 + 9 · 41 = 414 discs. The 415-th disc drawn will assure at least ten discs from a label.
74 Example (IMO 1964) Seventeen people correspond by mail with one another—each one with all the rest. In their letters
only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there at
least three people who write to each other about the same topic.
Solution: Choose a particular person of the group, say Charlie. He corresponds with sixteen others. By the Pigeonhole Principle,
Charlie must write to at least six of the people of one topic, say topic I. If any pair of these six people corresponds on topic I,
then Charlie and this pair do the trick, and we are done. Otherwise, these six correspond amongst themselves only on topics
II or III. Choose a particular person from this group of six, say Eric. By the Pigeonhole Principle, there must be three of the
five remaining that correspond with Eric in one of the topics, say topic II. If amongst these three there is a pair that corresponds
with each other on topic II, then Eric and this pair correspond on topic II, and we are done. Otherwise, these three people only
correspond with one another on topic III, and we are done again.
75 Example Given any seven distinct real numbers x1 , . . . x7 , prove that we can always find two, say a, b with
0<
a−b
1
<√ .
1 + ab
3
π
π
π π
< ak < . Divide the interval (− , ) into six non-overlapping subintervals of
2
2
2 2
π
equal length. By the Pigeonhole Principle, two of seven points will lie on the same interval, say ai < a j . Then 0 < a j − ai < .
6
Since the tangent increases in (−π /2, π /2), we obtain
Solution: Put xk = tan ak for ak satisfying −
0 < tan(a j − ai ) =
tan a j − tan ai
1
π
< tan = √ ,
1 + tan a j tan ai
6
3
as desired.
76 Example (Canadian Math Olympiad 1981) Let a1 , a2 , . . . , a7 be nonnegative real numbers with
a1 + a2 + . . . + a7 = 1.
If
M = max ak + ak+1 + ak+2 ,
1≤k≤5
determine the minimum possible value that M can take as the ak vary.
Solution: Since a1 ≤ a1 + a2 ≤ a1 + a2 + a3 and a7 ≤ a6 + a7 ≤ a5 + a6 + a7 we see that M also equals
max {a1 , a7 , a1 + a2 , a6 + a7 , ak + ak+1 + ak+2 }.
1≤k≤5
We are thus taking the maximum over nine quantities that sum 3(a1 + a2 + · · · + a7 ) = 3. These nine quantities then average
3/9 = 1/3. By the Pigeonhole Principle, one of these is ≥ 1/3, i.e. M ≥ 1/3. If a1 = a1 + a2 = a1 + a2 + a3 = a2 + a3 + a4 =
a3 + a4 + a5 = a4 + a5 + a6 = a5 + a6 + a7 = a7 = 1/3, we obtain the 7-tuple (a1 , a2 , a3 , a4 , a5 , a6 , a7 ) = (1/3, 0, 0, 1/3, 0, 0, 1/3),
which shows that M = 1/3.
Practice
Practice
15
77 Problem (AHSME 1991) A circular table has exactly sixty 84 Problem No matter which fifty five integers may be sechairs around it. There are N people seated at this table in such lected from
a way that the next person to be seated must sit next to some{1, 2, . . . , 100},
one. What is the smallest possible value of N?
prove that you must select some two that differ by 9, some two
that differ by 10, some two that differ by 12, and some two that
Answer: 20.
differ by 13, but that you need not have any two that differ by
11.
78 Problem Show that if any five points are all in, or on, a
square of√side 1, then some pair of them will be at most at
85 Problem Let mn + 1 different real numbers be given.
distance 2/2.
Prove that there is either an increasing sequence with at least
n + 1 members, or a decreasing sequence with at least m + 1
79 Problem (Eötvös, 1947) Prove that amongst six people in members.
a room there are at least three who know one another, or at least
three who do not know one another.
86 Problem If the points of the plane are coloured with three
colours, show that there will always exist two points of the
80 Problem Show that in any sum of non-negative real num- same colour which are one unit apart.
bers there is always one number which is at least the average
of the numbers and that there is always one member that it is
87 Problem Show that if the points of the plane are coloured
at most the average of the numbers.
with two colours, there will always exist an equilateral triangle with all its vertices of the same colour. There is, however, a
81 Problem We call a set “sum free” if no two elements of the colouring of the points of the plane with two colours for which
set add up to a third element of the set. What is the maximum no equilateral triangle of side 1 has all its vertices of the same
size of a sum free subset of {1, 2, . . . , 2n − 1}.
colour.
Hint: Observe that the set {n + 1, n + 2, . . . , 2n − 1} of n + 1 el- 88 Problem Let r , r , . . . , r , n > 1 be real numbers of abson
1 2
ements is sum free. Show that any subset with n + 2 elements lute value not exceeding 1 and whose sum is 0. Show that there
is not sum free.
is a non-empty proper subset whose sum is not more than 2/n
in size. Give an example in which any subsum has absolute
1
82 Problem (MMPC 1992) Suppose that the letters of the En.
value at least
glish alphabet are listed in an arbitrary order.
n−1
1. Prove that there must be four consecutive consonants.
2. Give a list to show that there need not be five consecutive consonants.
3. Suppose that all the letters are arranged in a circle. Prove
that there must be five consecutive consonants.
83 Problem (Stanford 1953) Bob has ten pockets and forty
four silver dollars. He wants to put his dollars into his pockets
so distributed that each pocket contains a different number of
dollars.
1. Can he do so?
89 Problem Let r1 , r2 , . . . , rn be real numbers in the interval
[0, 1]. Show that there are numbers εk , 1 ≤ k ≤ n, εk = −1, 0, 1
not all zero, such that
n
X
k=1
εk r k ≤
n
.
2n
90 Problem (USAMO, 1979) Nine mathematicians meet at
an international conference and discover that amongst any
three of them, at least two speak a common language. If
each of the mathematicians can speak at most three languages,
prove that there are at least three of the mathematicians who
can speak the same language.
2. Generalise the problem, considering p pockets and n
dollars. The problem is most interesting when
91 Problem (USAMO, 1982) In a party with 1982 persons,
amongst any group of four there is at least one person who
(p − 1)(p − 2)
knows each of the other three. What is the minimum number
.
n=
2
of people in the party who know everyone else?
Why?
16
92 Problem (USAMO, 1985) There are n people at a party.
Chapter 1
some three were sleeping simultaneously.
Prove that there are two people such that, of the remaining
n − 2 people, there are at least Tn/2U − 1 of them, each of
94 Problem Let Pn be a set of Ten!U + 1 points on the plane.
whom knows both or else knows neither of the two. Assume
Any two distinct points of Pn are joined by a straight line segthat “knowing” is a symmetrical relationship.
ment which is then coloured in one of n given colours. Show
that at least one monochromatic triangle is formed.
93 Problem (USAMO, 1986) During a certain lecture, each
of five mathematicians fell asleep exactly twice. For each pair
∞
X
of these mathematicians, there was some moment when both (Hint: e =
1/n!.)
were sleeping simultaneously. Prove that, at some moment,
n=0
Chapter
2
Divisibility
2.1 Divisibility
95 Definition If a 6= 0, b are integers, we say that a divides b if there is an integer c such that ac = b. We write this as a|b.
If a does not divide b we write a 6 |b. The following properties should be immediate to the reader.
96 Theorem
1. If a, b, c, m, n are integers with c|a, c|b, then c|(am + nb).
2. If x, y, z are integers with x|y, y|z then x|z.
Proof: There are integers s,t with sc = a,tc = b. Thus
am + nb = c(sm + tn),
giving c|(am + bn).
Also, there are integers u, v with xu = y, yv = z. Hence xuv = z, giving x|z.
It should be clear that if a|b and b 6= 0 then 1 ≤ |a| ≤ |b|.❑
97 Example Find all positive integers n for which
n + 1|n2 + 1.
Solution: n2 + 1 = n2 − 1 + 2 = (n − 1)(n + 1) + 2. This forces n + 1|2 and so n + 1 = 1 or n + 1 = 2. The choice n + 1 = 1 is
out since n ≥ 1, so that the only such n is n = 1.
98 Example If 7|3x + 2 prove that 7|(15x2 − 11x − 14.).
Solution: Observe that 15x2 − 11x − 14 = (3x + 2)(5x − 7). We have 7s = 3x + 2 for some integer s and so
15x2 − 11x − 14 = 7s(5x − 7),
giving the result.
Among every two consecutive integers there is an even one, among every three consecutive integers there is one divisible
by 3, etc.The following theorem goes further.
99 Theorem The product of n consecutive integers is divisible by n!.
17
18
Chapter 2
Proof: Assume first that all the consecutive integers m+1, m+2, . . . , m+n are positive. If this is so, the divisibility
by n! follows from the fact that binomial coefficients are integers:
m+n
(m + n)! (m + n)(m + n − 1) · · · (m + 1)
=
=
.
n
n!m!
n!
If one of the consecutive integers is 0, then the product of them is 0, and so there is nothing to prove. If all the n
consecutive integers are negative, we multiply by (−1)n , and see that the corresponding product is positive, and so
we apply the first result.❑
100 Example Prove that 6|n3 − n, for all integers n.
Solution: n3 − n = (n − 1)n(n + 1) is the product of 3 consecutive integers and hence is divisible by 3! = 6.
101 Example (Putnam 1966) Let 0 < a1 < a2 < . . . < amn+1 be mn + 1 integers. Prove that you can find either m + 1 of them
no one of which divides any other, or n + 1 of them, each dividing the following.
Solution: Let, for each 1 ≤ k ≤ mn+ 1, nk denote the length of the longest chain, starting with ak and each dividing the following
one, that can be selected from ak , ak+1 , . . . , amn+1 . If no nk is greater than n, then the are at least m + 1 nk ’s that are the same.
However, the integers ak corresponding to these nk ’s cannot divide each other, because ak |al implies that nk ≥ nl + 1.
102 Theorem If k|n then fk | fn .
Proof: Letting s = kn,t = n in the identity fs+t = fs−1 ft + fs ft+1 we obtain
f(k+1)n = fkn+n = fn−1 fkn + fn fkn+1 .
It is clear that if fn | fkn then fn | f(k+1)n . Since fn | fn·1 , the assertion follows.❑
Practice
n2 − 4, n2 + 8n + 7, n4 − 1, n2 − 2n?
107 Problem Prove that if n > 4 is composite, then n divides
(n − 1)!.
(Hint: Consider, separately, the cases when n is and is not a
perfect square.)
104 Problem Prove that n5 − 5n3 + 4n is always divisible by
108 Problem Prove that there is no prime triplet of the form
120.
p, p + 2, p + 4, except for 3, 5, 7.
103 Problem Given that 5|(n + 2), which of the following are
divisible by 5
109 Problem Prove that for n ∈ N, (n!)! is divisible by
105 Problem Prove that
(2m)!(3n)!
(m!)2 (n!)3
n!(n−1)!
110 Problem (AIME 1986) What is the largest positive inte-
is always an integer.
ger n for which
(n + 10)|(n3 + 100)?
106 Problem Demonstrate that for all integer values n,
n9 − 6n7 + 9n5 − 4n3
is divisible by 8640.
(Hint: x3 + y3 = (x + y)(x2 − xy + y2 ).)
111 Problem (Olimpíada matemática española, 1985) If n
is a positive integer, prove that (n + 1)(n + 2) · · · (2n) is divisible by 2n .
Division Algorithm
19
2.2 Division Algorithm
112 Theorem (Division Algorithm) If a, b are positive integers, then there are unique integers q, r such that a = bq + r, 0 ≤
r < b.
Proof: We use the Well-Ordering Principle. Consider the set S = {a − bk : k ∈ Z and a ≥ bk}. Then S is a
collection of nonnegative integers and S 6= ∅ as a − b · 0 ∈ S . By the Well-Ordering Principle, S has a least
element, say r. Now, there must be some q ∈ Z such that r = a − bq since r ∈ S . By construction, r ≥ 0. Let us
prove that r < b. For assume that r ≥ b. Then r > r − b = a − bq − b = a − (q + 1)b ≥ 0, since r − b ≥ 0. But then
a − (q + 1)b ∈ S and a − (q + 1)b < r which contradicts the fact that r is the smallest member of S . Thus we must
have 0 ≤ r < b. To show that r and q are unique, assume that bq1 + r1 = a = bq2 + r2 , 0 ≤ r1 < b, 0 ≤ r2 < b. Then
r2 − r1 = b(q1 − q2 ), that is b|(r2 − r1 ). But |r2 − r1 | < b, whence r2 = r1 . From this it also follows that q1 = q2 .
This completes the proof. ❑
It is quite plain that q = Ta/bU, where Ta/bU denotes the integral part of a/b.
It is important to realise that given an integer n > 0, the Division Algorithm makes a partition of all the integers according
to their remainder upon division by n. For example, every integer lies in one of the families 3k, 3k + 1 or 3k + 2 where k ∈ Z.
Observe that the family 3k + 2, k ∈ Z, is the same as the family 3k − 1, k ∈ Z. Thus
Z = A ∪ B ∪C
where
A = {. . . , −9, −6, −3, 0, 3, 6, 9, . . .}
is the family of integers of the form 3k, k ∈ Z,
B = {. . . − 8, −5, −2, 1, 4, 7, . . .}
is the family of integers of the form 3k + 1, k ∈ Z and
C = {. . . − 7, −4, −1, 2, 5, 8, . . .}
is the family of integers of the form 3k − 1, k ∈ Z.
113 Example (AHSME 1976) Let r be the remainder when 1059, 1417 and 2312 are divided by d > 1. Find the value of d − r.
Solution: By the Division Algorithm, 1059 = q1 d + r, 1417 = q2 d + r, 2312 = q3 d + r, for some integers q1 , q2 , q3 . From this,
358 = 1417 − 1059 = d(q2 − q1 ), 1253 = 2312 − 1059 = d(q3 − q1 ) and 895 = 2312 − 1417 = d(q3 − q2 ). Hence d|358 =
2 · 179, d|1253 = 7 · 179 and 7|895 = 5 · 179. Since d > 1, we conclude that d = 179. Thus (for example) 1059 = 5 · 179 + 164,
which means that r = 164. We conclude that d − r = 179 − 164 = 15.
114 Example Show that n2 + 23 is divisible by 24 for infinitely many n.
Solution: n2 + 23 = n2 − 1 + 24 = (n − 1)(n + 1) + 24. If we take n = 24k ± 1, k = 0, 1, 2, . . . , all these values make the expression
divisible by 24.
115 Definition A prime number p is a positive integer greater than 1 whose only positive divisors are 1 and p. If the integer
n > 1 is not prime, then we say that it is composite.
For example, 2, 3, 5, 7, 11, 13, 17, 19 are prime, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20 are composite. The number 1 is neither
a prime nor a composite.
116 Example Show that if p > 3 is a prime, then 24|(p2 − 1).
20
Chapter 2
Solution: By the Division Algorithm, integers come in one of six flavours: 6k, 6k ± 1, 6k ± 2 or 6k + 3. If p > 3 is a prime, then
p is of the form p = 6k ± 1 (the other choices are either divisible by 2 or 3). But (6k ± 1)2 − 1 = 36k2 ± 12k = 12k(3k − 1).
Since either k or 3k − 1 is even, 12k(3k − 1) is divisible by 24.
117 Example Prove that the square of any integer is of the form 4k or 4k + 1.
Solution: By the Division Algorithm, any integer comes in one of two flavours: 2a or 2a + 1. Squaring,
(2a)2 = 4a2 , (2a + 1)2 = 4(a2 + a) + 1)
and so the assertion follows.
118 Example Prove that no integer in the sequence
11, 111, 1111, 11111, . . .
is the square of an integer.
Solution: The square of any integer is of the form 4k or 4k + 1. All the numbers in this sequence are of the form 4k − 1, and so
they cannot be the square of any integer.
119 Example Show that from any three integers, one can always choose two so that a3 b − ab3 is divisible by 10.
Solution: It is clear that a3 b − ab3 = ab(a − b)(a + b) is always even, no matter which integers are substituted. If one of the
three integers is of the form 5k, then we are done. If not, we are choosing three integers that lie in the residue classes 5k ± 1 or
5k ± 2. Two of them must lie in one of these two groups, and so there must be two whose sum or whose difference is divisible
by 5. The assertion follows.
120 Example Prove that if 3|(a2 + b2 ), then 3|a and 3|b
Solution: Assume a = 3k ± 1 or b = 3m ± 1. Then a2 = 3x + 1, b2 = 3y + 1. But then a2 + b2 = 3t + 1 or a2 + b2 = 3s + 2, i.e.,
3 6 |(a2 + b2 ).
Practice
121 Problem Prove the following extension of the Division odd integers such that each is the sum of two squares greater
Algorithm: if a and b 6= 0 are integers, then there are unique than zero.
integers q and r such that a = qb + r, 0 ≤ r < |b|.
122 Problem Show that if a and b are positive integers, then
there are unique integers q and r, and ε = ±1 such that a =
b
b
qb + ε r, − < r ≤ .
2
2
123 Problem Show that the product of two numbers of the
126 Problem Let n > 1 be a positive integer. Prove that if
one of the numbers 2n − 1, 2n + 1 is prime, then the other is
composite.
127 Problem Prove that there are infinitely many integers n
such that 4n2 + 1 is divisible by both 13 and 5.
form 4k + 3 is of the form 4k + 1.
124 Problem Prove that the square of any odd integer leaves
128 Problem Prove that any integer n > 11 is the sum of two
positive composite numbers.
remainder 1 upon division by 8.
Hint: Think of n − 6 if n is even and n − 9 if n is odd.
125 Problem Demonstrate that there are no three consecutive
Some Algebraic Identities
21
129 Problem Prove that 3 never divides n2 + 1.
and also
x 6 |(y + 1) and (x + 1) 6 |(y + 1).
130 Problem Show the existence of infinitely many natural
numbers x, y such that x(x + 1)|y(y + 1) but
x 6 |y and (x + 1) 6 |y,
Hint: Try x = 36k + 14, y = (12k + 5)(18k + 7).
2.3 Some Algebraic Identities
In this section we present some examples whose solutions depend on the use of some elementary algebraic identities.
131 Example Find all the primes of the form n3 − 1, for integer n > 1.
Solution: n3 − 1 = (n − 1)(n2 + n + 1). If the expression were prime, since n2 + n + 1 is always greater than 1, we must have
n − 1 = 1, i.e. n = 2. Thus the only such prime is 7.
132 Example Prove that n4 + 4 is a prime only when n = 1 for n ∈ N.
Solution: Observe that
n4 + 4 = n4 + 4n2 + 4 − 4n2
= (n2 + 2)2 − (2n)2
= (n2 + 2 − 2n)(n2 + 2 + 2n)
= ((n − 1)2 + 1)((n + 1)2 + 1).
Each factor is greater than 1 for n > 1, and so n4 + 4 cannot be a prime.
133 Example Find all integers n ≥ 1 for which n4 + 4n is a prime.
Solution: The expression is only prime for n = 1. Clearly one must take n odd. For n ≥ 3 odd all the numbers below are integers:
n4 + 22n
=
n4 + 2n2 2n + 22n − 2n2 2n
=
=
(n2 + 2n )2 − n2(n+1)/2
(n2 + 2n + n2(n+1)/2 )(n2 + 2n − n2(n+1)/2 ).
€
Š2
It is easy to see that if n ≥ 3, each factor is greater than 1, so this number cannot be a prime.
134 Example Prove that for all n ∈ N , n2 divides the quantity
(n + 1)n − 1.
Solution: If n = 1 this is quite evident. Assume n > 1. By the Binomial Theorem,
n
(n + 1) − 1 =
n
X
k=1
n k
n,
k
and every term is divisible by n2 .
135 Example Prove that if p is an odd prime and if
a
= 1 + 1/2 + · · · + 1/(p − 1),
b
then p divides a.
22
Chapter 2
Solution: Arrange the sum as
1+
1
1
1
1
1
+ +
+···+
+
.
p−1 2 p−2
(p − 1)/2 (p + 1)/2
After summing consecutive pairs, the numerator of the resulting fractions is p. Each term in the denominator is < p. Since p is
a prime, the p on the numerator will not be thus cancelled out.
136 Example Prove that
xn − yn = (x − y)(xn−1 + xn−2 y + xn−3 y2 + · · · + xyn−2 + yn−1 )
Thus x − y always divides xn − yn .
Solution: We may assume that x 6= y, xy 6= 0, the result being otherwise trivial. In that case, the result follows at once from the
identity
n−1
X
an − 1
a 6= 1,
ak =
a−1
k=0
n
upon letting a = x/y and multiplying through by y .
☞ Without calculation we see that 8767
2345
− 81012345 is divisible by 666.
137 Example (Eőtvős 1899) Show that
2903n − 803n − 464n + 261n
is divisible by 1897 for all natural numbers n.
Solution: By the preceding problem, 2903n − 803n is divisible by 2903 − 803 = 2100 = 7 · 300 =, and 261n − 464n is divisible
by 261 − 464 = −203 = 7 · (−29). Thus the expression 2903n − 803n − 464n + 261n is divisible by 7. Also, 2903n − 464n is
divisible by 2903 − 464 = 9 · 271 and 261n − 803n is divisible by −542 = (−2)271. Thus the expression is also divisible by
271. Since 7 and 271 have no prime factors in common, we can conclude that the expression is divisible by 7 · 271 = 1897.
138 Example ((UM)2C4 1987) Given that 1002004008016032 has a prime factor p > 250000, find it.
Solution: If a = 103 , b = 2 then
1002004008016032 = a5 + a4 b + a3 b2 + a2 b3 + ab4 + b5 =
This last expression factorises as
a6 − b6
a−b
= (a + b)(a2 + ab + b2 )(a2 − ab + b2 )
=
=
1002 · 1002004 · 998004
4 · 4 · 1002 · 250501 · k,
where k < 250000. Therefore p = 250501.
139 Example (Grünert, 1856) If x, y, z, n are natural numbers n ≥ z, then the relation
xn + yn = zn
does not hold.
a6 − b6
.
a−b
Practice
23
Solution: It is clear that if the relation xn + yn = zn holds for natural numbers x, y, z then x < z and y < z. By symmetry, we may
suppose that x < y. So assume that xn + yn = zn and n ≥ z. Then
zn − yn = (z − y)(zn−1 + yzn−2 + · · · + yn−1 ) ≥ 1 · nxn−1 > xn ,
contrary to the assertion that xn + yn = zn . This establishes the assertion.
140 Example Prove that for n odd,
xn + yn = (x + y)(xn−1 − xn−2 y + xn−3 y2 − + − · · · + −xyn−2 + yn−1 ).
Thus if n is odd, x + y divides xn + yn .
Solution: This is evident by substituting −y for y in example 1.11 and observing that (−y)n = −yn for n odd.
141 Example Show that 1001 divides
11993 + 21993 + 31993 + · · · + 10001993 .
Solution: Follows at once from the previous problem, since each of 11993 + 10001993 , 21993 + 9991993 , . . . , 5001993 + 5011993 is
divisible by 1001.
142 Example (S250) Show that for any natural number n, there is another natural number x such that each term of the sequence
x
x + 1, xx + 1, xx + 1, . . .
is divisible by n.
Solution: It suffices to take x = 2n − 1.
143 Example Determine infinitely many pairs of integers (m, n) such that M and n share their prime factors and (m − 1, n − 1)
share their prime factors.
Solution: Take m = 2k − 1, n = (2k − 1)2 , k = 2, 3, . . .. Then m, n obviously share their prime factors and m − 1 = 2(2k−1 − 1)
shares its prime factors with n − 1 = 2k+1 (2k−1 − 1).
Practice
144 Problem Show that the integer
number
· · 2}
1| · ·{z
· · · · 1} − |2 ·{z
1...1
| {z }
91 ones
is composite.
145 Problem Prove that 199 + 299 + 399 + 499 is divisible by
5.
2n 1′ s
is the square of an integer.
148 Problem Let 0 ≤ a < b.
1. Prove that bn ((n + 1)a − nb) < an+1 .
2. Prove that for n = 1, 2, . . .,

146 Problem Show that if |ab| =
6 1, then a4 + 4b4 is compos-
ite.
147 Problem Demonstrate that for any natural number n, the
n 2′ s
1+
3. Show that
1
n
‹n

< 1+
1
n+1
‹n+1
n = 1, 2, . . . .
bn+1 − an+1
> (n + 1)a.
b−a
24
Chapter 2
4. Show that

158 Problem (Putnam, 1989) How many primes amongst
1
1+
n
‹n+1

1
> 1+
n+1
‹n+2
n = 1, 2, . . . .
149 Problem If a, b are positive integers, prove that
the positive integers, written as usual in base-ten are such that
their digits are alternating 1’s and 0’s, beginning and ending in
1?
159 Problem Find the least value achieved by 36k − 5k , k =
1, 2, . . . .
(a + 1/2)n + (b + 1/2)n
is an integer only for finitely many positive integers n.
160 Problem Find all the primes of the form n3 + 1.
150 Problem Prove that 100|1110 − 1.
161 Problem Find a closed formula for the product
2
151 Problem Let A and B be two natural numbers with the
n
P = (1 + 2)(1 + 22 )(1 + 22 ) · · · (1 + 22 ).
n
same number of digits, A > B. Suppose that A and B have
more than half of their digits on the sinistral side in common.
Prove that
1
A1/n − B1/n <
n
Use this to prove that for all positive integers n, 22 + 1 divides
for all n = 2, 3, 4, . . ..
pression
22
2n +1
− 2.
162 Problem Let a > 1 be a real number. Simplify the exÈ
È
√
√
a + 2 a − 1 + a − 2 a − 1.
152 Problem Demonstrate that every number in the sequence
· · 8} 9,
49, 4489, 444889, 44448889, . . . , 4| · ·{z
· · · · 4} 8| ·{z
n 4′ s
n−1 8′ s
163 Problem Let a, b, c, d be real numbers such that
a2 + b2 + c2 + d 2 = ab + bc + cd + da.
Prove that a = b = c = d.
is the square of an integer.
153 Problem (Polish Mathematical Olympiad) Prove that 164 Problem Let a, b, c be the lengths of the sides of a trianif n is an even natural number, then the number 13n + 6 is gle. Show that
divisible by 7.
3(ab + bc + ca) ≤ (a + b + c)2 ≤ 4(ab + bc + ca).
154 Problem Find, with proof, the unique square which is the
165 Problem (ITT, 1994) Let a, b, c, d be complex numbers
product of four consecutive odd numbers.
satisfying
155 Problem Prove that the number 22225555 + 55552222 is
divisible by 7.
Prove that a pair of the a, b, c, d must add up to 0.
(Hint: Consider
5555
2222
a + b + c + d = a3 + b3 + c3 + d 3 = 0.
5555
+4
166 Problem Prove that the product of four consecutive nat2222
+ 5555
2222
−4
2222
+4
5555
−4
.)
ural numbers is never a perfect square.
Hint: What is (n2 + n − 1)2 ?
156 Problem Prove that if an + 1, 1 < a ∈ N, is prime, then a
167 Problem Let k ≥ 2 be an integer. Show that if n is a
k
k
is even and n is a power of 2. Primes of the form 22 + 1 are positive integer, then n can be represented as the sum of n
successive odd numbers.
called Fermat primes.
157 Problem Prove that if an − 1, 1 < a ∈ N, is prime, then
168 Problem (Catalan) Prove that
n
a = 2 and n is a prime. Primes of the form 2 − 1 are called
Mersenne primes.
1−
1
1
1 1 1
+ − +···+
−
2 3 4
2n − 1 2n
Practice
25
equals
1
1
1
+
+···+ .
n+1 n+2
2n
write 4n + 1 as the sum of two squares, 4n + 1 = x2 + y2 where
x and y are expressed in terms of a and b.
Conversely, show that if 4n + 1 = x2 + y2 , then n is the sum
of two triangular numbers.
169 Problem (IMO, 1979) If a, b are natural numbers such
that
1 1 1
1
1
a
= 1− + − +···−
+
,
b
2 3 4
1318 1319
prove that 1979|a.
173 Problem (Polish Mathematical Olympiad) Prove
that
amongst ten successive natural numbers, there are always at
least one and at most four numbers that are not divisible by
any of the numbers 2, 3, 5, 7.
170 Problem (Polish Mathematical Olympiad) A triangu-
lar number is one of the form 1 + 2 + . . . + n, n ∈ N. Prove 174 Problem Show that if k is odd,
that none of the digits 2, 4, 7, 9 can be the last digit of a triangular number.
1+2+···+n
171 Problem Demonstrate that there are infinitely many
square triangular numbers.
divides
1k + 2k + · · · + nk .
172 Problem (Putnam, 1975) Supposing that an integer n is
the sum of two triangular numbers,
a2 + a b2 + b
+
,
n=
2
2
175 Problem Are there five consecutive positive integers
such that the sum of the first four, each raised to the fourth
power, equals the fifth raised to the fourth power?
Chapter
3
Congruences. Zn
3.1 Congruences
The notation a ≡ b mod n is due to Gauß, and it means that n|(a − b). It also indicates that a and b leave the same remainder
upon division by n. For example, −8 ≡ −1 ≡ 6 ≡ 13 mod 7. Since n|(a − b) implies that ∃k ∈ Z such that nk = a − b, we
deduce that a ≡ b mod n if and only if there is an integer k such that a = b + nk.
We start by mentioning some simple properties of congruences.
176 Lemma Let a, b, c, d, m ∈ Z, k ∈ with a ≡ b mod m and c ≡ d mod m. Then
1. a + c ≡ b + d mod m
2. a − c ≡ b − d mod m
3. ac ≡ bd mod m
4. ak ≡ bk mod m
5. If f is a polynomial with integral coefficients then f (a) ≡ f (b) mod m.
Proof: As a ≡ b mod m and c ≡ d mod m, we can find k1 , k2 ∈ Z with a = b + k1 m and c = d + k2 m. Thus
a ± c = b ± d + m(k1 ± k2 ) and ac = bd + m(k2 b + k1 d). These equalities give (1), (2) and (3). Property (4) follows
by successive application of (3), and (5) follows from (4). ❑
Congruences mod 9 can sometimes be used to check multiplications. For example 875961 · 2753 6= 2410520633. For if
this were true then
(8 + 7 + 5 + 9 + 6 + 1)(2 + 7 + 5 + 3) ≡ 2 + 4 + 1 + 0 + 5 + 2 + 0 + 6 + 3 + 3
But this says that 0 · 8 ≡ 8 mod 9, which is patently false.
177 Example Find the remainder when 61987 is divided by 37.
Solution: 62 ≡ −1 mod 37. Thus 61987 ≡ 6 · 61986 ≡ 6(62 )993 ≡ 6(−1)993 ≡ −6 ≡ 31 mod 37.
178 Example Prove that 7 divides 32n+1 + 2n+2 for all natural numbers n.
Solution: Observe that 32n+1 ≡ 3 · 9n ≡ 3 · 2n mod 7 and 2n+2 ≡ 4 · 2n mod 7. Hence
32n+1 + 2n+2 ≡ 7 · 2n ≡ 0
for all natural numbers n.
26
mod 7,
mod 9.
Congruences
27
179 Example Prove the following result of Euler: 641|(232 + 1).
Solution: Observe that 641 = 27 · 5 + 1 = 24 + 54 . Hence 27 · 5 ≡ −1 mod 641 and 54 ≡ −24 mod 641. Now, 27 · 5 ≡ −1
mod 641 yields 54 · 228 = (5 · 27 )4 ≡ (−1)4 ≡ 1 mod 641. This last congruence and 54 ≡ −24 mod 641 yield −24 · 228 ≡ 1
mod 641, which means that 641|(232 + 1).
180 Example Find the perfect squares mod 13.
Solution: First observe that we only have to square all the numbers up to 6, because r2 ≡ (13 − r)2 mod 13. Squaring the
nonnegative integers up to 6, we obtain 02 ≡ 0, 12 ≡ 1, 22 ≡ 4, 32 ≡ 9, 42 ≡ 3, 52 ≡ 12, 62 ≡ 10 mod 13. Therefore the perfect
squares mod 13 are 0, 1, 4, 9, 3, 12, and 10.
181 Example Prove that there are no integers with x2 − 5y2 = 2.
Solution: If x2 = 2 − 5y2 , then x2 ≡ 2 mod 5. But 2 is not a perfect square mod 5.
182 Example Prove that 7|(22225555 + 55552222 ).
Solution: 2222 ≡ 3 mod 7, 5555 ≡ 4 mod 7 and 35 ≡ 5 mod 7. Now 22225555 + 55552222 ≡ 35555 + 42222 ≡ (35 )1111 +
(42 )1111 ≡ 51111 − 51111 ≡ 0 mod 7.
7
183 Example Find the units digit of 77 .
7
Solution: We must find 77 mod 10. Now, 72 ≡ −1 mod 10, and so 73 ≡ 72 · 7 ≡ −7 ≡ 3 mod 10 and 74 ≡ (72 )2 ≡ 1
mod 10. Also, 72 ≡ 1 mod 4 and so 77 ≡ (72 )3 · 7 ≡ 3 mod 4, which means that there is an integer t such that 77 = 3 + 4t.
Upon assembling all this,
7
77 ≡ 74t+3 ≡ (74 )t · 73 ≡ 1t · 3 ≡ 3
mod 10.
Thus the last digit is 3.
184 Example Prove that every year, including any leap year, has at least one Friday 13-th.
Solution: It is enough to prove that each year has a Sunday the 1st. Now, the first day of a month in each year falls in one of the
following days:
Month
Day of the year
mod 7
January
1
1
February
32
4
March
60 or 61
4 or 5
April
91 or 92
0 or 1
May
121 or122
2 or 3
June
152 or 153
5 or 6
July
182 or183
0 or 1
August
213 or 214
3 or 4
September 244 or 245
6 or 0
October
274 or 275
1 or 2
November 305 or 306
4 or 5
December 335 or 336
6 or 0
(The above table means that, depending on whether the year is a leap year or not, that March 1st is the 50th or 51st day of the
year, etc.) Now, each remainder class modulo 7 is represented in the third column, thus each year, whether leap or not, has at
least one Sunday the 1st.
28
Chapter 3
185 Example Find infinitely many integers n such that 2n + 27 is divisible by 7.
Solution: Observe that 21 ≡ 2, 22 ≡ 4, 23 ≡ 1, 24 ≡ 2, 25 ≡ 4, 26 ≡ 1 mod 7 and so 23k ≡ 1 mod 3 for all positive integers k.
Hence 23k + 27 ≡ 1 + 27 ≡ 0 mod 7 for all positive integers k. This produces the infinitely many values sought.
186 Example Are there positive integers x, y such that x3 = 2y + 15?
Solution: No. The perfect cubes mod 7 are 0, 1, and 6. Now, every power of 2 is congruent to 1, 2, or 4 mod 7. Thus
2y + 15 ≡ 2, 3, or 5 mod 7. This is an impossibility.
187 Example Prove that 2k − 5, k = 0, 1, 2, . . . never leaves remainder 1 when divided by 7.
Solution: 21 ≡ 2, 22 ≡ 4, 23 ≡ 1 mod 7, and this cycle of three repeats. Thus 2k − 5 can leave only remainders 3, 4, or 6 upon
division by 7.
188 Example (AIME, 1994) The increasing sequence
3, 15, 24, 48, . . . ,
consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994-th term of
the sequence is divided by 1000?
Solution: We want 3|n2 − 1 = (n − 1)(n + 1). Since 3 is prime, this requires n = 3k + 1 or n = 3k − 1, k = 1, 2, 3, . . .. The
sequence 3k + 1, k = 1, 2, . . . produces the terms n2 − 1 = (3k + 1)2 − 1 which are the terms at even places of the sequence of
3, 15, 24, 48, . . .. The sequence 3k − 1, k = 1, 2, . . . produces the terms n2 − 1 = (3k − 1)2 − 1 which are the terms at odd places
of the sequence 3, 15, 24, 48, . . .. We must find the 997th term of the sequence 3k + 1, k = 1, 2, . . .. Finally, the term sought is
(3(997) + 1)2 − 1 ≡ (3(−3) + 1)2 − 1 ≡ 82 − 1 ≡ 63 mod 1000. The remainder sought is 63.
189 Example (USAMO, 1979) Determine all nonnegative integral solutions
(n1 , n2 , . . . , n14 )
if any, apart from permutations, of the Diophantine equation
n41 + n42 + · · · + n414 = 1599.
Solution: There are no such solutions. All perfect fourth powers mod 16 are ≡ 0 or 1 mod 16. This means that
n41 + · · · + n414
can be at most 14 mod 16. But 1599 ≡ 15 mod 16.
190 Example (Putnam, 1986) What is the units digit of
T
1020000
U?
10100 + 3
200 100
Solution: Set a−3 = 10
200
X
(−1)k
Since
k=0
20000
. Then [(10
199
100
)/10
200
+3] = [(a−3)
k=0
X
200
200
= 0, (3)199
= −3199 . As a ≡ 3 mod 10,
(−1)k
k
k
k=0
199 X
k=0
199
X
200 199−k
200
a
(−3)k ≡ 3199
(−1)k
≡ −3199 ≡ 3 mod 10.
k
k
k=0
199 X 200
1 X 200 200−k
/a] = [
a
(−3)k ] =
a199−k (−3)k .
k
k
a
k=0
Congruences
29
191 Example Prove that for any a, b, c ∈ Z, n ∈ N, n > 3, there is an integer k such that n 6 |(k + a), n 6 |(k + b), n 6 |(k + c).
Solution: The integers a, b, c belong to at most three different residue classes mod n. Since n > 3, we have more than three
distinct residue classes. Thus there must be a residue class, say k for which −k 6≡ a, −k 6≡ b, −k 6≡ c, mod n. This solves the
problem.
192 Example (Putnam, 1973) Let a1 , a2 , . . . , a2n+1 be a set of integers such that if any one of them is removed, the remaining
ones can be divided into two sets of n integers with equal sums. Prove that a1 = a2 = . . . = a2n+1 .
Solution: As the sum of the 2n integers remaining is always even, no matter which of the ak be taken, all the ak must have the
same parity. The property stated in the problem is now shared by ak /2 or (ak − 1)/2, depending on whether they are all even, or
all odd. Thus they are all congruent mod 4. Continuing in this manner we arrive at the conclusion that the ak are all congruent
mod 2k for every k, and this may only happen if they are all equal.
193 Example Prove that
(kn)! ≡ 0
n−1
Y
(n + r)
mod
r=0
if n, k ∈ N, n ≥ k ≥ 2.
Solution: (kn)! = M(n − 1)!n(n + 1) · · · (2n − 1) for some integer M ≥ 1. The assertion follows.
194 Example Let
n!! = n! (1/2! − 1/3! + · · · + (−1)n /n!) .
Prove that for all n ∈ N, n > 3,
Solution: We have
n!! ≡ n!
n! − n!!
=
=
=
mod (n − 1).
n(n − 1)(n − 2)!(1 − 1/2!
+ ·€· · + (−1)n−1 /(n − 1)! + (−1)n /n!)
Š
(n − 1) m + (−1)n−1 n/(n − 1) + (−1)n /(n − 1)
(n − 1) (m + (−1)n ) ,
where M is an integer, since (n − 2)! is divisible by k!, k ≤ n − 2.
195 Example Prove that
6n+2
X
k=0
6n + 2 k
3 ≡ 0, 23n+1 , −23n+1
2k
mod 23n+2
when n is of the form 2k, 4k + 3 or 4k + 1 respectively.
Solution: Using the Binomial Theorem,
2S := 2
3n+1
X
k=0
√
√
6n + 2 k
3 = (1 + 3)6n+2 + (1 − 3)6n+2 .
2k
√
√
Also, if n is odd, with a = 2 + 3, b = 2 − 3,
1 3n+1
(a
+ b3n+1 )
2
=
3n + 1
2 3n + 1
X
r=0
2r
23n+1−2r 3r .
≡ 3(3n+1)/2 mod 4
≡ (−1)(n−1)/2 mod 4.
30
Chapter 3
As 2S = 23n+1 (a3n+1 + b3n+1 ), we have, for odd n,
S ≡ (−1)(n−1)/2 23n+1
If n is even,
1 3n+1
(a
+ b3n+1 )
2
X
=
2r≤3n
mod 23n+3 .
3n + 1 2r+1 3n−2r
2
3
2r + 1
≡ 2(6n + 1)33n mod 8
≡ 4n + 2 mod 8.
So for even n, S ≡ 23n+2 2n + 1 mod 23n+4 .
Practice
196 Problem Find the number of all n, 1 ≤ n ≤ 25 such that
n2 + 15n + 122 is divisible by 6.
207 Problem Prove that the sum of the decimal digits of a
perfect square cannot be equal to 1991.
(Hint: n2 + 15n + 122 ≡ n2 + 3n + 2 = (n + 1)(n + 2) mod 6.) 208 Problem Prove that
197 Problem (AIME 1983) Let an = 6n + 8n . Determine the
n
n
7|42 + 22 + 1
remainder when a83 is divided by 49.
for all natural numbers n.
198 Problem (P OLISH M ATHEMATICAL O LYMPIAD ) What
digits should be put instead of x and y in 30x0y03 in order to
209 Problem Prove that 5 never divides
give a number divisible by 13?
3
3
n
X
3
2
199 Problem Prove that if 9|(a + b + c ), then 3|abc, for
integers a, b, c.
k=0
200 Problem Describe all integers n such that 10|n10 + 1.
201 Problem Prove that if
a − b, a2 − b2 , a3 − b3 , a4 − b4 , . . .
3k
2n + 1
.
2k + 1
n
n
− [ ] is divisip
p
210 Problem Prove that if p is a prime,
ble by p, for all n ≥ p.
211 Problem How many perfect squares are there mod 2n ?
are all integers, then a and b must also be integers.
202 Problem Find the last digit of 3100 .
203 Problem (AHSME 1992) What is the size of the largest
subset S of {1, 2, . . . , 50} such that no pair of distinct elements
of S has a sum divisible by 7?
212 Problem Prove that every non-multiple of 3 is a perfect
power of 2 mod 3n .
213 Problem Find the last two digits of 3100 .
equation x2 − 7y = 3.
214 Problem (USAMO, 1986) What is the smallest integer
n > 1, for which the root-mean-square of the first n positive
integers is an integer?
205 Problem Prove that if 7|a2 + b2 then 7|a and 7|b.
Note.
204 Problem Prove that there are no integer solutions to the
The root mean square of n numbers a1 , a2 , . . . , an is defined to be

206 Problem Prove that there are no integers with
800000007 = x2 + y2 + z2 .
a21 + a22 + · · · + a2n
n
‹1/2
.
Divisibility Tests
31
215 Problem Find all integers a, b, c, a > 1 and all prime 219 Problem Let x and ai , i = 0, 1, . . . , k be arbitrary integers.
numbers p, q, r which satisfy the equation
Prove that
k
X
pa = qb + rc
(a, b, c, p, q, r need not necessarily be different).
ai (x2 + 1)3i
i=0
216 Problem Show that the number 16 is a perfect 8-th power
mod p for any prime p.
is divisible by x2 ± x + 1 if and only if
217 Problem (IMO, 1975) Let a1 , a2 , a3 , . . . be an increasing
by x2 ± x + 1.
k
X
(−1)i ai is divisible
i=0
sequence of positive integers. Prove that for every s ≥ 1
there are infinitely many am that can be written in the form 220 Problem ((UM)2C9 1992) If x, y, z are positive integers
with
am = xas + yat with positive integers x and y and t > s.
n
xn + yn = zn
2
218 Problem For each integer n > 1, prove that n −n +n−1
is divisible by (n − 1)2 .
for an odd integer n ≥ 3, prove that z cannot be a prime-power.
3.2 Divisibility Tests
Working base-ten, we have an ample number of rules of divisibility. The most famous one is perhaps the following.
221 Theorem (Casting-out 9’s) A natural number n is divisible by 9 if and only if the sum of it digits is divisible by 9.
Proof: Let n = ak 10k + ak−1 10k−1 + · · · + a1 10 + a0 be the base-10 expansion of n. As 10 ≡ 1 mod 9, we have
10 j ≡ 1 mod 9. It follows that n = ak 10k + · · · + a1 10 + a0 ≡ ak + · · · + a1 + a0 , whence the theorem. ❑
222 Example (AHSME, 1992) The two-digit integers from 19 to 92 are written consecutively in order to form the integer
192021222324 · · · 89909192.
What is the largest power of 3 that divides this number?
Solution: By the casting-out-nines rule, this number is divisible by 9 if and only if
19 + 20 + 21 + · · · + 92 = 372 · 3
is. Therefore, the number is divisible by 3 but not by 9.
223 Example (IMO, 1975) When 44444444 is written in decimal notation, the sum of its digits is A. Let B be the sum of the
digits of A. Find the sum of the digits of B. (A and B are written in decimal notation.)
Solution: We have 4444 ≡ 7 mod 9, and hence 44443 ≡ 73 ≡ 1 mod 9. Thus 44444444 = 44443(1481) · 4444 ≡ 1 · 7 ≡ 7 mod 9.
Let C be the sum of the digits of B.
By the casting-out 9’s rule, 7 ≡ 44444444 ≡ A ≡ B ≡ C mod 9. Now, 4444 log10 4444 < 4444 log10 104 = 17776. This means
that 44444444 has at most 17776 digits, so the sum of the digits of 44444444 is at most 9 · 17776 = 159984, whence A ≤ 159984.
Amongst all natural numbers ≤ 159984 the one with maximal digit sum is 99999, so it follows that B ≤ 45. Of all the natural
numbers ≤ 45, 39 has the largest digital sum, namely 12. Thus the sum of the digits of B is at most 12. But since C ≡ 7 mod 9,
it follows that C = 7.
A criterion for divisibility by 11 can be established similarly. For let n = ak 10k + ak−1 10k−1 + · · · + a1 10 + a0 . As 10 ≡ −1
mod 11, we have 10 j ≡ (−1) j mod 11. Therefore n ≡ (−1)k ak + (−1)k−1 ak−1 + · · · − a1 + a0 mod 11, that is, n is divisible by
11 if and only if the alternating sum of its digits is divisible by 11. For example, 912282219 ≡ 9−1+2−2+8−2+2−1+9 ≡ 7
mod 11 and so 912282219 is not divisible by 11, whereas 8924310064539 ≡ 8 − 9 + 2 − 4 + 3 − 1 + 0 − 0 + 6 − 4 + 4 − 3 + 9 ≡ 0
mod 11, and so 8924310064539 is divisible by 11.
32
Chapter 3
224 Example (Putnam, 1952) Let
f (x) =
n
X
ak xn−k
k=0
be a polynomial of degree n with integral coefficients. If a0 , an and f (1) are all odd, prove that f (x) = 0 has no rational roots.
Solution: Suppose that f (a/b) = 0, where a and b are relatively prime integers. Then 0 = bn f (a/b) = a0 bn + a1 bn−1 a + · · · +
an−1 ban−1 + an an . By the relative primality of a and b it follows that a|a0 , b|an , whence a and b are both odd. Hence
a0 bn + aa bn−1 a + · · · + an−1 ban−1 + an an ≡ a0 + a1 + · · · + an = f (1) ≡ 1
mod 2,
but this contradicts that a/b is a root of f .
Practice
225 Problem (AHSME 1991) An n-digit integer is cute if its
n digits are an arrangement of the set {1, 2, . . . , n} and its first k
digits form an integer that is divisible by k for all k, 1 ≤ k ≤ n.
For example, 321 is a cute three-digit number because 1 divides 3, 2 divides 32, and 3 divides 321. How many cute sixdigit integers are there?
piles. What is the smallest amount of coconuts that could have
been in the original pile?
Answer: 15621
230 Problem Prove that a number which consists of 3n iden-
Answer: 2.
tical digits is divisible by 3n . For example, 111 111 111 is
divisible by 9.
226 Problem How many ways are there to roll two distin-
231 Problem ((UM)2C8 1991) Suppose that a0 , a1 , . . . an are
guishable dice to yield a sum that is divisible by three?
integers with an 6= 0, and let
p(x) = a0 + a1 x + · · · + an xn .
Answer: 12.
227 Problem Prove that a number is divisible by 2k , k ∈ N if
and only if the number formed by its last k digits is divisible
by 2k . Test whether
90908766123456789999872
is divisible by 8.
228 Problem An old receipt has faded. It reads 88 chickens
at the total of $x4.2y, where x and y are unreadable digits. How
much did each chicken cost?
Suppose that x0 is a rational number such that p(x0 ) = 0. Show
that if 1 ≤ k ≤ n, then
ak x0 + ak+1 x02 + · · · + an xn−k+1
is an integer.
232 Problem 1953 digits are written in a circular order. Prove
that if the 1953-digit numbers obtained when we read these
digits in dextrogyral sense beginning with one of the digits is
divisible by 27, then if we read these digits in the same direction beginning with any other digit, the new 1953-digit number
is also divisible by 27.
Answer: 73 cents.
233 Problem (Lagrange) Prove that
229 Problem Five sailors plan to divide a pile of coconuts
fn+60 ≡ fn mod 10.
amongst themselves in the morning. During the night, one of
them wakes up and decides to take his share. After throwing a
coconut to a monkey to make the division come out even, he Thus the last digit of a Fibonacci number recurs in cycles of
takes one fifth of the pile and goes back to sleep. The other length 60.
four sailors do likewise, one after the other, each throwing
a coconut to the monkey and taking one fifth of the remain- 234 Problem Prove that
ing pile. In the morning the five sailors throw a coconut to
2
f2n+1 ≡ fn+1
mod fn2 .
the monkey and divide the remaining coconuts into five equal
Complete Residues
33
3.3 Complete Residues
The following concept will play a central role in our study of integers.
235 Definition If a ≡ b mod n then b is called a residue of a modulo n. A set a1 , a2 , . . . an is called a complete residue system
modulo n if for every integer b there is exactly one index j such that b ≡ a j mod n.
It is clear that given any finite set of integers, this set will form a complete set of residues modulo n if and only if the
set has n members and every member of the set is incongruent modulo n. For example, the set A = {0, 1, 2, 3, 4, 5} forms
a complete set of residues mod 6, since any integer x is congruent to one and only one member of A . Notice that the set
B = {−40, 6, 7, 15, 22, 35} forms a complete residue set mod 6, but the set C = {−3, −2, −1, 1, 2, 3} does not, as −3 ≡ 3
mod 6.
+3
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
Table 3.1: Addition Table for Z3
+6
0
1
2
3
4
5
0
0
1
2
3
4
5
1
1
2
3
4
5
0
2
2
3
4
5
0
1
3
3
4
5
0
1
2
4
4
5
0
1
2
3
5
5
0
1
2
3
4
Table 3.2: Addition Table for Z6
Tied up with the concept of complete residues is that of Zn . As an example, let us take n = 3. We now let 0 represent all
those integers that are divisible by 3, 1 represent all those integers that leave remainder 1 upon division by 3, and 2 all those
integers that leave remainder 2 upon division by 3, and consider the set Z3 = {0, 1, 2}. We define addition in Z3 as follows.
Given a, b ∈ Z3 we consider a + b mod 3. Now, there is c ∈ {0, 1, 2} such that a + b ≡ c mod 3. We then define a +3 b to be
equal to c. Table 3.3 contains all the possible additions.
We observe that Z3 together with the operation +3 as given in Table 3.3 satisfies the following properties:
1. The element 0 ∈ Z3 is an identity element for Z3 , i.e. 0 satisfies 0 +3 a = a +3 0 = a for all a ∈ Z3
2. Every element a ∈ Z3 has an additive inverse b, i.e., an element such that a +3 b = b +3 a = 0. We denote the additive
inverse of a by −a. In Z3 we note that −0 = 0, −1 = 2, −2 = 1.
3. The operation addition in Z3 is associative, that is, for all a, b, c ∈ Z3 we have a +3 (b +3 c) = (a +3 b) +3 c.
We then say that < Z3 , +3 > forms a group and we call it the group of residues under addition mod 3.
Similarly we define < Zn , +n >, as the group of residues under addition mod n. As a further example we present the
addition table for < Z6 , +6 > on Table (1.2). We will explore later the multiplicative structure of Zn .
Practice
236 Problem Construct the addition tables for Z8 and Z9 .
237 Problem How many distinct ordered pairs (a, b) 6= (0, 0)
are in Z12 such that a +12 b = 0?
Chapter
4
Unique Factorisation
4.1 GCD and LCM
If a, b ∈ Z, not both zero, the largest positive integer that divides both a, b is called the greatest common divisor of a and b. This
is denoted by (a, b) or sometimes by gcd(a, b). Thus if d|a and d|b then d|(a, b), because any common divisor of a and b must
divide the largest common divisor of a and b. For example, (68, −6) = 2, gcd(1998, 1999) = 1.
If (a, b) = 1, we say that a and b are relatively prime or coprime. Thus if a, b are relatively prime, then they have no factor
greater than 1 in common.
If a, b are integers, not both zero, the smallest positive integer that is a multiple of a, b is called the least common multiple
of a and b. This is denoted by [a, b]. We see then that if a|c and if b|c, then [a, b]|c, since c is a common multiple of both a and
b, it must be divisible by the smallest common multiple of a and b.
The most important theorem related to gcd’s is probably the following.
238 Theorem (Bachet-Bezout Theorem) The greatest common divisor of any two integers a, b can be written as a linear
combination of a and b, i.e., there are integers x, y with
(a, b) = ax + by.
Proof: Let A = {ax + by|ax + by > 0, x, y ∈ Z}. Clearly one of ±a, ±b is in A , as both a, b are not zero. By the
Well Ordering Principle, A has a smallest element, say d. Therefore, there are x0 , y0 such that d = ax0 + by0 . We
prove that d = (a, b). To do this we prove that d|a, d|b and that if t|a,t|b, then t|d.
We first prove that d|a. By the Division Algorithm, we can find integers q, r, 0 ≤ r < d such that a = dq + r. Then
r = a − dq = a(1 − qx0 ) − by0 .
If r > 0, then r ∈ A is smaller than the smaller element of A , namely d, a contradiction. Thus r = 0. This entails
dq = a, i.e. d|a. We can similarly prove that d|b.
Assume that t|a,t|b. Then a = tm, b = tn for integers m, n. Hence d = ax0 + bx0 = t(mx0 + ny0 ), that is, t|d. The
theorem is thus proved. ❑
☞ It is clear that any linear combination of a, b is divisible by (a, b).
239 Lemma (Euclid’s Lemma) If a|bc and if (a, b) = 1, then a|c.
Proof: As (a, b) = 1, by the Bachet-Bezout Theorem, there are integers x, y with ax + by = 1. Since a|bc, there is
an integer s with as = bc. Then c = c · 1 = cax + cby = cax + asy. From this it follows that a|c, as wanted.❑
34
GCD and LCM
35
240 Theorem If (a, b) = d, then
a b
( , ) = 1.
d d
Proof: By the Bachet-Bezout Theorem, there are integers x, y such that ax + by = d. But then (a/d)x + (b/d)y = 1,
and a/d, b/d are integers. But this is a linear combination of a/d, b/d and so (a/d, b/d) divides this linear
combination, i.e., divides 1. We conclude that (a/d, b/d) = 1.❑
241 Theorem Let c be a positive integer. Then
(ca, cb) = c(a, b).
Proof: Let d1 = (ca, cb) and d2 = (a, b). We prove that d1 |cd2 and cd2 |d1 . As d2 |a and d2 |b, then cd2 |ca, cd2 |cb.
Thus cd2 is a common divisor of ca and cb and hence d1 |cd2 . By the Bachet-Bezout Theorem we can find integers
x, y with d1 = acx + bcy = c(ax + by). But ax + by is a linear combination of a, b and so it is divisible by d2 . There
is an integer s then such that sd2 = ax + by. It follows that d1 = csd2 , i.e., cd2 |d1 . ❑
☞ It follows similarly that (ca, cb) = |c|(a, b) for any non-zero integer c.
242 Lemma For nonzero integers a, b, c,
(a, bc) = (a, (a, b)c).
Proof: Since (a, (a, b)c) divides (a, b)c it divides bc. Thus gcd(a, (a, b)c) divides a and bc and hence gcd(a, (a, b)c)| gcd(a, bc).
On the other hand, (a, bc) divides a and bc, hence it divides ac and bc. Therefore (a, bc) divides (ac, bc) = c(a, b).
In conclusion, (a, bc) divides a and c(a, b) and so it divides (a, (a, b)c). This finishes the proof.❑
243 Theorem (a2 , b2 ) = (a, b)2 .
Proof: Assume that (m, n) = 1. Using the preceding lemma twice,
(m2 , n2 ) = (m2 , (m2 , n)n) = (m2 , (n, (m, n)m)n).
As (m, n) = 1, this last quantity equals (m2 , n). Using the preceding problem again,
(m2 , n) = (n, (m, n)m) = 1.
Thus (m, n) = 1 implies (m2 , n2 ) = 1.
By Theorem 240,
and hence

b
a
,
(a, b) (a, b)
‹
b2
a2
,
2
(a, b) (a, b)2
= 1,
= 1.
By Theorem 241, upon multiplying by (a, b)2 , we deduce
(a2 , b2 ) = (a, b)2 ,
which is what we wanted.❑
244 Example Let (a, b) = 1. Prove that (a + b, a2 − ab + b2 ) = 1 or 3.
Solution: Let d = (a + b, a2 − ab + b2 ). Now d divides
(a + b)2 − a2 + ab − b2 = 3ab.
Hence d divides 3b(a + b) − 3ab = 3b2 . Similarly, d|3a2 . But then d|(3a2 , 3b2 ) = 3(a2 , b2 ) = 3(a, b)2 = 3.
36
Chapter 4
245 Example Let a, a 6= 1, m, n be positive integers. Prove that
(am − 1, an − 1) = a(m,n) − 1.
Solution: Set d = (m, n), sd = m,td = n. Then am − 1 = (ad )s − 1 is divisible by ad − 1 and similarly, an − 1 is divisible by
ad − 1. Thus (ad − 1)|(am − 1, an − 1). Now, by the Bachet-Bezout Theorem there are integers x, y with mx + ny = d. Notice that
x and y must have opposite signs (they cannot obviously be both negative, since then d would be negative. They cannot both be
positive because then d ≥ m + n, when in fact we have d ≤ m, d ≤ n). So, assume without loss of generality that x > 0, y ≤ 0.
Set t = (am − 1, an − 1). Then t|(amx − 1) and t|(a−ny − 1). Hence, t|((amx − 1) − ad (a−ny − 1)) = ad − 1. The assertion is
established.
246 Example (IMO, 1959) Prove that the fraction
21n + 4
is irreducible for every natural number n.
14n + 3
Solution: 2(21n + 4) − 3(14n + 3) = −1. Thus the numerator and the denominator have no common factor greater than 1.
247 Example (AIME, 1985) The numbers in the sequence
101, 104, 109, 116, . . .
are of the form an = 100 + n2 , n = 1, 2, . . .. For each n let dn = (an , an+1 ). Find max dn .
n≥1
Solution: We have the following: dn = (100 + n2 , 100 + (n + 1)2 ) = (100 + n2 , 100 + n2 + 2n + 1) = (100 + n2 , 2n + 1). Thus
dn |(2(100+n2 )−n(2n+1)) = 200−n. Therefore dn |(2(200−n)+(2n+1)) = 401. This means that dn |401 for all n. Could it be
that large? The answer is yes, for let n = 200, then a200 = 100 + 2002 = 100(401) and a201 = 100 + 2012 = 40501 = 101(401).
Thus max dn = 401.
n≥1
248 Example Prove that if m and n are natural numbers and m is odd, then (2m − 1, 2n + 1) = 1.
Solution: Let d = (2m − 1, 2n + 1). It follows that d must be an odd number, and 2m − 1 = kd, 2n + 1 = ld, for some natural
numbers k, l. Therefore, 2mn = (kd + 1)n = td + 1, where t =
n−1
X
j=0
n n− j n− j−1
k d
. In the same manner, 2mn = (ld − 1)m =
j
ud − 1, where we have used the fact that m is odd. As td + 1 = ud − 1, we must have d|2, whence d = 1.
249 Example Prove that there are arbitrarily long arithmetic progressions in which the terms are pairwise relatively prime.
Solution: The numbers km! + 1, k = 1, 2, . . . , m form an arithmetic progression of length m and common difference m!. Suppose
that d|(lm! + 1), d|(sm! + 1), 1 ≤ l < s ≤ m. Then d|(s(lm! + 1) − l(sm! + 1)) = (s − l) < m. Thus 1 ≤ d < m and so, d|m!. But
then d|(sm! + 1 − sm!) = 1. This means that any two terms of this progression are coprime.
250 Example Prove that any two consecutive Fibonacci numbers are relatively prime.
Solution: Let d = ( fn , fn+1 ). As fn+1 − fn = fn−1 and d divides the sinistral side of this equality, d| fn−1 . Thus d|( fn − fn−1 ) =
fn−2 . Iterating on this process we deduce that d| f1 = 1 and so d = 1.
Aliter: By Cassini’s Identity fn−1 fn+1 − fn2 = (−1)n . Thus d|(−1)n , i.e., d = 1.
251 Example Prove that
( fm , fn ) = f(n,m) .
Solution: Set d = ( fn , fm ), c = f(m,n) , a = (m, n). We will prove that c|d and d|c.
GCD and LCM
37
Since a|m and a|n, fa | fm and fa | fn by Theorem 102. Thus
fa |( fm , fm ),
i.e., c|d.
Now, by the Bachet-Bezout Theorem, there are integers x, y such that xm + yn = a. Observe that x, y cannot be both negative,
otherwise a would be negative. As a|n, a|m we have a ≤ n, a ≤ m. They cannot be both positive since then a = xm + yn ≥ m + n,
a contradiction. Thus they are of opposite signs, and we assume without loss of generality that x ≤ 0, y > 0.
Observe that
fyn = fa−xm = fa−1 f−xm + fa f−xm+1
upon using the identity
fs+t = fs−1 ft + fs ft+1
of Theorem 50. As n|yn, m|(−xm), we have that fn | fyn , fm | f−xm . This implies that ( fn , fm )| fyn and ( fn , fm )| f−xm . Hence
( fn , fm )| fa f−xm+1 .
We saw earlier that ( fn , fm )| f−xm . If it were the case that
( fn , fm )| f−xm+1 ,
then ( fn , fm ) would be dividing two consecutive Fibonacci numbers, a contradiction to the preceding problem in the case when
( fn , fm ) > 1. The case = 1 is a triviality. Therefore ( fn , fm )| fa , which is what we wanted to prove.
252 Example Prove that no odd Fibonacci number is ever divisible by 17.
Solution: Let d = (17, fn ), which obviously must be odd. Then (17, fn ) = (34, fn ) = ( f9 , fn ) = f(9,n) = f1 , f3 or f9 . This means
that d = (17, fn ) = 1, 2 or 34. This forces d = 1.
253 Example The Catalan number of order n is defined as
Cn =
2n
1
.
n+1 n
Prove that Cn is an integer for all natural numbers n.
Solution: By the binomial absorption identity,
2n + 1 2n
2n + 1
=
.
n+1 n
n+1
Since 2n + 1 and n + 1 are relatively prime, and since the dextral side is an integer, it must be the case that n + 1 divides
254 Example Let n be a natural number. Find the greatest common divisor of
Solution: Since
2n
2n
2n
,
,...,
.
1
3
2n − 1
n
X
k=1
2n
= 22n−1 ,
2k − 1
2n
.
n
38
Chapter 4
2n
the gcd must be of the form 2 . Since the gcd must divide
= 2n, we see that it has divide 2l+1 , where l is the largest
1
power of 2 that divides n. We claim that 2l+1 divides all of them. We may write n = 2l m, where M is odd. Now,
a
2l+1 m 2l+1 m − 1
2l+1 m
=
.
2k − 1
2k − 1
2k − 2
But 2k − 1 6 |2l+1 for k > 1. This establishes the claim.
255 Example Let any fifty one integers be taken from amongst the numbers 1, 2, . . . , 100. Show that there are two that are
relatively prime.
Solution: Arrange the 100 integers into the 50 sets
{1, 2}, {3, 4}, {5, 6} . . . , {99, 100}.
Since we are choosing fifty one integers, there must be two that will lie in the same set. Those two are relatively prime, as
consecutive integers are relatively prime.
256 Example Prove that any natural number n > 6 can be written as the sum of two integers greater than 1, each of the
summands being relatively prime.
Solution: If n is odd, we may choose a = 2, b = n − 2. If n is even, then is either of the form 4k or 4k + 2. If n = 4k, then take
a = 2k + 1, b = 2k − 1. These two are clearly relatively prime (why?). If n = 4k + 2, k > 1 take a = 2k + 3, b = 2k − 1.
257 Example How many positive integers ≤ 1260 are relatively prime to 1260?
Solution: As 1260 = 22 · 32 · 5 · 7, the problem amounts to finding those numbers less than 1260 which are not divisible by
2, 3, 5, or 7. Let A denote the set of integers ≤ 1260 which are multiples of 2, B the set of multiples of 3, etc. By the
Inclusion-Exclusion Principle,
|A ∪ B ∪C ∪ D| =
=
|A| + |B| + |C| + |D|
−|A ∩ B| − |A ∩C| − |A ∩ D|
−|B ∩C| − |B ∩ D| − |C ∩ D|
+|A ∩ B ∩C| + |A ∩ B ∩ D| + |A ∩C ∩ D|
+|B ∩C ∩ D| − |A ∩ B ∩C ∩ D|
630 + 420 + 252 + 180 − 210 − 126 − 90 − 84
−60 − 36 + 42 + 30 + 18 + 12 − 6 = 972.
The number of integers sought is then 1260 − 972 = 288.
Practice
258 Problem Show that
260 Problem Find two positive integers a, b such that
a2 + b2 = 85113, and lcm (a, b) = 1764.
(a, b)[a, b] = ab
for all natural numbers a, b.
259 Problem Find lcm (23!41!, 29!37!).
261 Problem Find a, b ∈ N with (a, b) = 12, [a, b] = 432.
262 Problem Prove that (a, b)n = (an , bn ) for all natural num-
bers n.
Primes
39
263 Problem Let a ∈ N. Find, with proof, all b ∈ N such that
(2b − 1)|(2a + 1).
n
268 Problem Let Fn = 22 + 1 be the n-th Fermat number.
Find (Fn , Fm ).
269 Problem Find the greatest common divisor of the se-
264 Problem Show that (n3 + 3n + 1, 7n3 + 18n2 − n − 2) = 1.
quence
16n + 10n − 1, n = 1, 2, . . . .
265 Problem Let the integers an , bn be defined by the relation
√
√
an + bn 2 = (1 + 2)n , n ∈ N.
270 Problem Demonstrate that (n! + 1, (n + 1)! + 1) = 1.
271 Problem Prove that any natural number n > 17 can be
Prove that gcd(an , bn ) = 1 ∀ n.
266 Problem Prove or disprove the following two proposi-
written as n = a + b + c where a, b, c are pairwise relatively
prime natural numbers each exceeding 1.
tions:
1. If a, b ∈ N, a < b, then in any set of b consecutive integers there are two whose product is divisible by ab.
(Hint: Consider n mod 12. Write two of the summands in the
form 6k + s and the third summand as a constant.)
2. If a, b, c, ∈ N, a < b < c, then in any set of c consecu- 272 Problem Prove that there are no positive integers a, b, n >
tive integers there are three whose product is divisible 1 with
(an − bn )|(an + bn ).
by abc.
267 Problem Let n, k, n ≥ k > 0 be integers. Prove that the
greatest common divisor of the numbers
n
n+1
n+k
,
,...,
k
k
k
273 Problem Prove that the binomial coefficients have the
following hexagonal property:

gcd
equals
is 1.

gcd
(Hint: Prove
k
X
(−1)
j=0
j
k
j
‹
‹
n−1
n
n+1
,
,
k−1
k+1
k
n−1
n+1
n
,
,
k
k+1
k−1
.
274 Problem (Putnam, 1974) Call a set of integers conspir-
n+ j
= (−1)k .)
k
atorial if no three of them are pairwise relatively prime. What
is the largest number of elements in any conspiratorial subset
of the integers 1 through 16?
4.2 Primes
Recall that a prime number is a positive integer greater than 1 whose only positive divisors are itself and 1. Clearly 2 is the only
even prime and so 2 and 3 are the only consecutive integers which are prime. An integer different from 1 which is not prime is
called composite. It is clear that if n > 1 is composite then we can write n as n = ab, 1 < a ≤ b < n, a, b ∈ N.
275 Theorem If n > 1, then n is divisible by at least one prime.
Proof: Since n > 1, it has at least one divisor > 1. By the Well Ordering Principle, n must have a least positive
divisor greater than 1, say q. We claim that q is prime. For if not then we can write q as q = ab, 1 < a ≤ b < q. But
then a is a divisor of n greater than 1 and smaller than q, which contradicts the minimality of q.❑
276 Theorem (Euclid) There are infinitely many primes.
Proof: Let p1 , p2 , . . . pk be a list of primes. Construct the integer
n = p1 p2 · · · pk + 1.
40
Chapter 4
This integer is greater than 1 and so by the preceding problem, it must have a prime divisor p. Observe that p must
be different from any of p1 , p2 , . . . , pk since n leaves remainder 1 upon division by any of the pi . Thus we have
shown that no finite list of primes exhausts the set of primes, i.e., that the set of primes is infinite.❑
277 Lemma The product of two numbers of the form 4k + 1 is again of that form.
Proof: (4a + 1)(4b + 1) = 4(4ab + a + b) + 1.❑
278 Theorem There are infinitely many primes of the form 4n + 3.
Proof: Any prime either equals 2, or is of the form 4k ± 1. We will show that the collection of primes of the form
4k − 1 is inexhaustible. Let
{p1 , p2 , . . . pn }
be any finite collection of primes of the form 4k − 1. Construct the number
N = 4p1 p2 · · · pn − 1.
Since each pk is ≥ 3, N ≥ 11. Observe that N is not divisible by any of the primes in our collection. Now either N
is a prime, in which case it is a prime of the form 4k − 1 not on the list, or it is a product of primes. In the latter
case, all of the prime factors of N cannot be of the form 4k + 1, for the product of any two primes of this form is
again of this form, in view of the preceding problem. Thus N must be divisible by some prime of the form 4k − 1
not on the list. We have thus shown that given any finite list of primes of the form 4k − 1 we can always construct
an integer which is divisible by some prime of the form 4k − 1 not on that list. The assertion follows. ❑
279 Example Prove that there are arbitrarily long strings that do not contain a prime number.
Solution: Let k ∈ N, k ≥ 2. Then each of the numbers
k! + 2, . . . , k! + k
is composite.
280 Theorem If the positive integer n is composite, then it must have a prime factor p with p ≤
√
n.
√ √
√
Proof: Suppose that n = ab, 1√
< a ≤ b < n. If both a and b are > n, then
√ n = ab > n n = n, a contradiction.
Thus n has a factor 6= 1 and ≤ n, and hence a prime factor, which is ≤ n. ❑
281 Example Find the number of prime numbers ≤ 100.
√
Solution: Observe that 100 = 10. By the preceding theorem, all the composite numbers in the range 10 ≤ n ≤ 100 have
a prime factor amongst 2, 3, 5, or 7. Let Am denote the multiples of M which are ≤ 100. Then |A2 | = 50, |A3 | = 33, |A5 | =
20, |A7 | = 14, |A6 | = 16, |A10 | = 10, |A14 | = 7, |A15 | = 6, |A21 | = 4, |A35 | = 2, |A30 | = 3, |A42 | = 2, |A70 | = 1, |A105 | = 0, |A210 | = 0.
Thus the number of primes ≤ 100 is
=
=
=
=
100 − ( number of composites ≤ 1) − 1
4 + 100 − multiples of 2, 3, 5, or 7 ≤ 100 − 1
4 + 100 − (50 + 33 + 20 + 14) + (16 + 10 + 7 + 6 + 4 + 2)
−(3 + 2 + 1 + 0) − 0 − 1
25,
where we have subtracted the 1, because 1 is neither prime nor composite.
282 Lemma If p is a prime,
p
is divisible by p for all 0 < k < p.
k
Practice
41
Proof:
p(p − 1) · · · (p − k + 1)
p
=
k!
k
yields
k!
p
= p(p − 1) · · · (p − k + 1),
k
p
p
. Now, as k < p, p 6 |k!. By Euclid’s Lemma, it must be the case that p|
.❑
whence p|k!
k
k
283 Example Prove that if p is a prime, then p divides 2 p − 2.
Solution: By the Binomial Theorem:
p
p
p
2 − 2 = (1 + 1) − 2 =
+
+···+
,
1
2
p−1
p
p
=
0
assertion.
as
p
p
= 1. By the preceding lemma, p divides each of the terms on the dextral side of the above. This establishes the
p
Practice
284 Problem Prove that there are infinitely many primes of
the form 6n + 5.
285 Problem Use the preceding problem to show that there
are infinitely many primes p such that p − 2 is not a prime.
1).
4. Prove that 42|n7 − n, n ∈ Z.
5. Prove that 30|n5 − n, n ∈ Z.
288 Problem Let p be an odd prime and let (a, b) = 1. Prove
286 Problem If p and q are consecutive odd primes, prove
that the prime factorisation of p + q has at least three (not necessarily distinct) primes.
that

ap + bp
a + b,
a+b
‹
divides p.
1. Let p be a prime and let n ∈ N. Prove, by 289 Problem Prove that 3, 5, 7 is the only prime triplet of the
form p, p + 2, p + 4.
induction on n, that p|(n p − n).
287 Problem
2. Extend this result to all n ∈ Z.
3. Prove Fermat’s Little Theorem: if p 6 |n, then p|(n p−1 −
290 Problem Let n > 2. Prove that if one of the numbers
2n − 1 and 2n + 1 is prime, then the other is composite.
4.3 Fundamental Theorem of Arithmetic
Consider the integer 1332. It is clearly divisible by 2 and so we obtain 1332 = 2 · 666. Now, 666 is clearly divisible by 6, and
so 1332 = 2 · 2 · 3 · 111. Finally, 111 is also divisible by 3 and so we obtain 1332 = 2 · 2 · 3 · 3 · 37. We cannot further decompose
1332 as a product of positive integers greater than 1, as all 2, 3, 37 are prime. We will show now that such decomposition is
always possible for a positive integer greater than 1.
291 Theorem Every integer greater than 1 is a product of prime numbers.
Proof: Let n > 1. If n is a prime, then we have nothing to prove. Assume that n is composite and let q1 be its least
proper divisor. By Theorem 4.5, q1 is a prime. Set n = q1 n1 , 1 < n1 < n. If n1 is a prime, then we arrived at the
result. Otherwise, assume that n1 is composite, and let q2 be its least prime divisor, as guaranteed by Theorem 4.5.
42
Chapter 4
We can write then n = q1 q2 n2 , 1 < n2 < n1 < n. Continuing the argument, we arrive at a chain n > n1 > n2 · · · > 1,
and this process must stop before n steps, as n is a positive integer. Eventually we then have n = q1 q2 · · · qs . ❑
We may arrange the prime factorisation obtained in the preceding Theorem as follows,
a
n = pa11 pa22 · · · pk k , a1 > 0, a2 > 0, . . . , ak > 0,
p1 < p2 < · · · < pk ,
where the p j are primes. We call the preceding factorisation of n, the canonical factorisation of n. For example 23 32 52 73 is the
canonical factorisation of 617400.
292 Theorem (Fundamental Theorem of Arithmetic) Every integer > 1 can be represented as a product of primes in only
one way, apart from the order of the factors.
Proof: We prove that a positive integer greater than 1 can only have one canonical factorisation. Assume that
n = pa11 pa22 · · · pas s = qb11 qb22 · · · qtbt
are two canonical factorisations of n. By Euclid’s Lemma (example 1.2) we conclude that every p must be a q and
every q must be a p. This implies that s = t. Also, from p1 < p2 < · · · < ps and q1 < q2 < · · · < qt we conclude that
p j = q j , 1 ≤ j ≤ s.
b
If a j > b j for some j then, upon dividing by p j j , we obtain
a −b j
pa11 pa22 · · · p j j
b
b
j−1
j+1
· · · pas s = pb11 pb22 · · · p j−1
p j+1
· · · pbs s ,
which is impossible, as the sinistral side is divisible by p j and the dextral side is not. Similarly, the alternative
a j < b j for some j is ruled out and so a j = b j for all j. This finishes the proof. ❑
It is easily seen, by the Fundamental Theorem of Arithmetic, that if a has the prime factorisation a = pa11 pa22 · · · pann and b
has the prime factorisation b = pb11 pb22 · · · pbnn , (it may be the case that some of the ak and some of the bk are zero) then
min(a1 ,b1 ) min(a2 ,b2 )
min(an ,bn )
p2
· · · pn
.
(4.1)
max(a1 ,b1 ) max(a2 ,b2 )
max(an ,bn )
p2
· · · pn
.
(4.2)
(a, b) = p1
and also
[a, b] = p1
Since x + y = max(x, y) + min(x, y), it clearly follows that
ab = (a, b)[a, b].
293 Example Prove that
√
2 is irrational.
√
Solution: Assume that 2 = a/b with relatively prime natural numbers a, b. Then 2b2 = a2 . The sinistral side of this last
equality has an odd number of prime factors (including repetitions), whereas the dextral side has an even number of prime
factors. This contradicts the Fundamental Theorem of Arithmetic.
294 Example Prove that if the polynomial
p(x) = a0 xn + a1 xn−1 + · · · + an−1 x + an
with integral coefficients assumes the value 7 for four integral values of x, then it cannot take the value 14 for any integral value
of x.
Fundamental Theorem of Arithmetic
43
Solution: First observe that the integer 7 can be decomposed into at most three different integer factors 7 = −7(1)(−1). Assume
that p(ak ) − 7 = 0 for distinct ak , 1 ≤ k ≤ 4. Then
p(x) − 7 = (x − a1 )(x − a2 )(x − a3 )(x − a4 )q(x)
for a polynomial q with integer coefficients. Assume that there is an integer M with p(m) = 14. Then
7 = p(m) − 7 = (m − a1 )(m − a2 )(m − a3 )(m − a4 )q(m).
Since the factors m − ak are all distinct, we have decomposed the integer 7 into at least four different factors. This is impossible,
by the Fundamental Theorem of Arithmetic.
295 Example Prove that the product of three consecutive integers is never a perfect power (i.e., a perfect square or a perfect
cube, etc.).
Solution: Let the integer be (n−1)n(n+1) = (n2 −1)n. Since n2 −1 and n are relatively prime, by the Fundamental Theorem of
Arithmetic, n2 − 1 is a perfect kth power (k ≥ 2) and n is also a perfect kth power. But then, n2 − 1 and n2 would be consecutive
perfect kth powers, sheer nonsense.
296 Example Prove that m5 + 3m4 n − 5m3 n2 − 15m2 n3 + 4mn4 + 12n5 is never equal to 33.
Solution: Observe that
m5 + 3m4 n − 5m3 n2 − 15m2 n3 + 4mn4 + 12n5
= (m − 2n)(m − n)(m + n)(m + 2n)(m + 3n).
Now, 33 can be decomposed as the product of at most four different integers 33 = (−11)(3)(1)(−1). If n 6= 0, the factors in the
above product are all different. They cannot be multiply to 33, by the Fundamental Theorem of Arithmetic, as 33 is the product
of 4 different factors and the expression above is the product of 5 different factors for n 6= 0.. If n = 0, the product of the factors
is m5 , and 33 is clearly not a fifth power.
297 Example Prove that the sum
S = 1/2 + 1/3 + 1/4 + · · · + 1/n
is never an integer.
Solution: Let k be the largest integer such that 2k ≤ n, and P the product of all the odd natural numbers not exceeding n. The
1
number 2k−1 PS is a sum, all whose terms, except for 2k−1 P k , are integers.
2
298 Example Prove that there is exactly one natural number n for with 28 + 211 + 2n is a perfect square.
Solution: If k2 = 28 + 211 + 2n = 2304 + 2n = 482 + 2n , then k2 − 482 = (k − 48)(k + 48) = 2n . By unique factorisation,
k − 48 = 2s , k + 48 = 2t , s +t = n. But then 2t − 2s = 96 = 3 · 25 or 2s (2t−s − 1) = 3 · 25 . By unique factorisation, s = 5,t − s = 2,
giving s + t = n = 12.
299 Example Prove that in any set of 33 distinct integers with prime factors amongst {5, 7, 11, 13, 23}, there must be two
whose product is a square.
Solution: Any number in our set is going to have the form
5a 7b 11c 13d 23 f .
Thus to each number in the set, we associate a vector (a, b, c, d, f ). These vectors come in 32 different flavours, according to
the parity of the components. For example (even, odd, odd, even, odd) is one such class. Since we have 33 integers, two (at
least) will have the same parity in their exponents, and the product of these two will be a square.
44
Chapter 4
300 Example (IMO, 1985) Given a set M of 1985 distinct positive integers, none with a prime factor greater than 26, prove
that M contains a subset of four distinct elements whose product is the fourth power of an integer.
Solution: Any number in our set is going to be of the form
2a 3b 5c 7d 11 f 13g 17h 19 j 23k .
Thus if we gather 513 of these numbers, we will have two different ones whose product is a square.
Start weeding out squares. Since we have 1985 > 513 numbers, we can find a pair of distinct a1 , b1 such that a1 b1 = c21 .
Delete this pair. From the 1983 integers remaining, we can find a pair of distinct a2 , b2 such that a2 b2 = c22 . Delete this pair.
From the 1981 integers remaining, we can find a pair a3 , b3 such that a3 b3 = c23 . We can continue this operation as long as
we have at least 513 integers. Thus we may perform this operation n + 1 times, were n is the largest positive integer such that
1985 − 2n ≥ 513, i.e., n = 736. Therefore, we are able to gather 737 pairs ak , bk such that ak bk = c2k . Now, the 737 numbers
ck have all their prime factors smaller than 26, and since 737 > 513, we may find two distinct cm say ci and c j , i 6= j, such that
ci c j = a2 , a perfect square. But then ci c j = a2 implies that ai bi a j b j = a4 , a fourth power. Thus we have found four distinct
numbers in our set whose product is a fourth power.
301 Example Let any fifty one integers be taken from amongst the numbers 1, 2, . . . , 100. Show that there must be one that
divides some other.
Solution: Any of the fifty one integers can be written in the form 2a m, where m is odd. Since there are only fifty odd integers
between 1 and 100, there are only fifty possibilities for m. Thus two (at least) of the integers chosen must share the same odd
part, and thus the smaller will divide the larger.
302 Example (USAMO 1972) Prove that
(a, b, c)2
[a, b, c]2
=
.
[a, b][b, c][c, a] (a, b)(b, c)(c, a)
Solution: Put
a=
Y
α
pk k , b =
Y
β
pk k , c =
Y
γ
pkk ,
with primes pk . The assertion is equivalent to showing
2 max(αk , βk , γk ) − max(αk , βk ) − max(αk , γk ) − max(βk , γk )
= 2 min(αk , βk , γk ) − min(αk , βk ) − min(αk , γk ) − min(βk , γk ).
By symmetry, we may assume, without loss of generality, that αk ≥ βk ≥ γk . The equation to be established reduces thus to the
identity
2αk − αk − αk − βk = 2γk − βk − γk − γk .
303 Example Prove that n = 24 is the largest natural number divisible by all integral a, 1 ≤ a ≤
√
n.
√
√
Solution: Suppose n is divisible by all the integers ≤ n. Let p1 = 2, p2 = 3, . . . , pl be all the primes ≤ n, and let k j be the
√
√
√
k
k +1
k +1
unique integers such that p j j ≤ n < p j j . Clearly nl/2 < p1k1 +1 p2k2 +1 · · · pl l . Let lcm(1, 2, 3, . . . , T nU − 1, T nU) = K.
k +1
Clearly then K = pk11 pk22 · · · pl l . Hence p1k1 +1 p2k2 +1 · · · pl l ≤ K 2 and thus nl/2 < K 2 . By hypothesis, n must be divisible by K
and so K ≤ n. Consequently, nl/2 < n2 . This implies that l < 4 and so n < 49. By inspection, we see that the only valid values
for n are n = 2, 4, 6, 8, 12, 24.
k
304 Example (Irving Kaplansky) A positive integer n has the property that for 0 < l < m < n,
S = l + (l + 1) + . . . + m
is never divisible by n. Prove that this is possible if and only if n is a power of 2.
Practice
45
Solution: Set n = s2k with s odd. If s = 1, 2S = (l + m)(m − l + 1), which has one factor even and one factor odd, cannot be
divisible by 2n = 2k+1 , since, its even factor is less than 2n. But if s > 1, then S is divisible by n, with 0 < l < m < n, if we take
m = (s + 2k+1 − 1)/2
and
l=
1 + m − 2k+1 , s > 2k+1 ,
1 + m − s,
s < 2k+1 .
305 Example Let 0 < a1 < a2 < · · · < ak ≤ n, where k > T
n+1
U, be integers. Prove that
2
a1 + a j = ar
is soluble.
Solution: The k − 1 positive integers ai − a1 , 2 ≤ i ≤ k, are clearly distinct. These, together with the k given distinct a’s, give
2k − 1 > n positive integers, each not greater than n. Hence, at least one of the integers is common to both sets, so that at least
once ar − a1 = a j .
The sequence Tn/2U + 1, Tn/2U + 2, . . . , n, shows that for k = T(n + 1)/2U the result is false.
306 Example Let 0 < a1 < a2 < · · · < an ≤ 2n be integers such that the least common multiple of any two exceeds 2n. Prove
that a1 > T
2n
U.
3
Solution: It is clear that no one of the numbers can divide another (otherwise we would have an lcm ≤ 2n). Hence, writing
ak = 2tk Ak , Ak odd, we see that all the Ak are different. Since there are n of them, they coincide in some order with the set of all
positive odd numbers less than 2n.
Now, consider a1 = 2t1 A1 . If a1 ≤ T2n/3U, then 3a1 = 2t1 3A1 ≤ 2n, and 3A1 < 2n. Since 3A1 would then be an odd number
< 2n, 3A1 = A j for some j, and a j = 2t j 3A1 . Thus either [a1 , a j ] = 2t1 3A1 = 3a1 ≤ 2n, or [a1 , a j ] = 2t j 3A1 = a j ≤ 2n. These
contradictions establish the assertion.
307 Example (Putnam, 1980) Derive a formula for the number of quadruples (a, b, c, d) such that
3r 7s = [a, b, c] = [b, c, d] = [c, d, a] = [d, a, b].
Solution: By unique factorisation, each of a, b, c, d must be of the form 3m 7n , 0 ≤ m ≤ r, 0 ≤ n ≤ s. Moreover,
M must equal
4 2
r = 6r2 ways
r for at least two of the four numbers, and n must equal s for at least two of the four numbers. There are
2
4
of choosing exactly two of the four numbers to have exponent r,
r = 4r ways of choosing exactly three to have exponent
3
4
= 1 of choosing the four to have exponent r. Thus there is a total of 1 + 4r + 6r2 of choosing at least two of the
r and
4
four numbers to have exponent r. Similarly, there are 1 + 4s + 6s2 ways of choosing at least two of the four numbers to have
exponent s. The required formula is thus
(1 + 4r + 6r2 )(1 + 4s + 6s2 ).
Practice
308 Problem Prove that log10 7 is irrational.
309 Problem Prove that
log 3
log 2
46
Chapter 4
is irrational.
310 Problem Find the smallest positive integer such that n/2
is a square and n/3 is a cube.
319 Problem Let 2 = p1 , 3 = p2 , . . . be the primes in their natural order and suppose that n ≥ 10 and that 1 < j < n. Set
N1 = p1 p2 · · · p j−1 − 1, N2 = 2p1 p2 · · · p j−1 − 1, . . .
and
20
inclusive,
are not perfect squares, perfect cubes, or perfect fifth powers?
Np j = p j p1 p2 · · · p j−1 − 1
311 Problem How many integers from 1 to 10
Prove
1. Each pi , j ≤ i ≤ n, divides at most one of the Npk , 1 ≤
k≤ j
312 Problem Prove that the sum
1/3 + 1/5 + 1/7 + · · · + 1/(2n + 1)
2. There is a j, 1 < j < n, for which p j > n − j + 1.
is never an integer.
3. Let s be the smallest j for which p j > n − j + 1. There
is a t, 1 ≤ t ≤ ps , such that all of p1 , . . . pn fail to divide
t p1 p2 · · · ps−1 − 1, and hence pn+1 < p1 p2 · · · ps .
(Hint: Look at the largest power of 3 ≤ n).
313 Problem Find min 36k − 5k .
4. The s above is > 4 and so ps−1 − 2 ≥ s and p1 p2 · · · ps <
ps+1 · · · pn .
(Hint: Why is 36k − 1 − 5k 6= 0?)
5. (Bonse’s Inequality) For n ≥ 4, p2n+1 < p1 · · · pn .
k≥1
320 Problem Prove that 30 is the only integer n with the foltriples (a, b, c) of positive integers for which [a, b] = lowing property: if 1 ≤ t ≤ n and (t, n) = 1, then t is prime.
1000, [b, c] = [a, c] = 2000.
314 Problem (AIME 1987) Find the number of ordered
1. For which positive integers n is there a finite set Sn of n distinct positive integers such that the geometric mean of any subset of Sn is
an integer?
321 Problem (USAMO 1984)
315 Problem Find the number of ways of factoring 1332
as the product of two positive relatively prime factors each
greater than 1. Factorisations differing in order are considered
the same.
2. Is there an infinite set S of distinct positive integers such
that the geometric mean of any finite subset of S is an
integer.
Answer: 3.
316 Problem Let p1 , p2 , . . . , pt be different primes and
1. (Putnam 1955) Prove that there is no
a1 , a2 , . . . at be natural numbers. Find the number of ways of 322 Problem
a1 a2
triplet
of
integers
(a, b, c), except for (a, b, c) = (0, 0, 0)
at
factoring p1 p2 · · · pt as the product of two positive relatively
for
which
√
√
prime factors each greater than 1. Factorisations differing in
a
+
b
2
+
c
3 = 0.
order are considered the same.
Answer: 2t−1 − 1.
a
a
a
b
b
b
317 Problem Let n = p11 p22 · · · pt t and m = p11 p22 · · · pt t , the
p’s being different primes. Find the number of the common
factors of m and n.
Answer:
t
Y
(1 + min(ak , bk )).
k=1
318 Problem (USAMO 1973) Show that the cube roots of
three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.
2. (Putnam 1980) Prove that there exist integers a, b, c, not
all zero and each of absolute value less than a million,
such that
√
√
|a + b 2 + c 3| < 10−11 .
3. (Putnam 1980) Let a, b, c be integers, not all zero and
each of absolute value less than a million. Prove that
√
√
|a + b 2 + c 3| > 10−21 .
323 Problem (Eőtvős 1906) Let a1 , a2 , . . . , an be any permutation of the numbers 1, 2, . . . , n. Prove that if n is odd, the
product
(a1 − 1)(a2 − 2) · · · (an − n)
is an even number.
Practice
324 Problem Prove that from any sequence formed by arranging in a certain way the numbers from 1 to 101, it is always
possible to choose 11 numbers (which must not necessarily be
consecutive members of the sequence) which form an increasing or a decreasing sequence.
325 Problem Prove that from any fifty two integers it is al-
47
326 Problem Prove that from any one hundred integers it is
always possible to choose several numbers (or perhaps, one
number) whose sum is divisible by 100.
327 Problem Given n numbers x1 , x2 , . . . , xn each of which is
equal to ±1, prove that if
x1 x2 + x2 x3 + · · · + xn x1 = 0,
ways to choose two, whose sum, or else, whose difference, is
divisible by 100.
then n is a multiple of 4.
Chapter
5
Linear Diophantine Equations
5.1 Euclidean Algorithm
We now examine a procedure that avoids factorising two integers in order to obtain their greatest common divisor. It is called the
Euclidean Algorithm and it is described as follows. Let a, b be positive integers. After using the Division Algorithm repeatedly,
we find the sequence of equalities
a
b
r2
..
.
=
=
=
..
.
bq1 + r2 ,
r2 q2 + r3
r3 q3 + r4
..
.
0 < r2 < b,
0 < r3 < r2 ,
0 < r4 < r3 ,
..
.
rn−2
rn−1
=
=
rn−1 qn−1 + rn
rn qn .
0 < rn < rn−1 ,
(5.1)
The sequence of remainders will eventually reach a rn+1 which will be zero, since b, r2 , r3 , . . . is a monotonically decreasing
sequence of integers, and cannot contain more than b positive terms.
The Euclidean Algorithm rests on the fact, to be proved below, that (a, b) = (b, r2 ) = (r2 , r3 ) = · · · = (rn−1 , rn ) = rn .
328 Theorem Prove that if a, b, n are positive integers, then
(a, b) = (a + nb, b).
Proof: Set d = (a, b), c = (a + nb, b). As d|a, d|b, it follows that d|(a + nb). Thus d is a common divisor of both
(a + nb) and b. This implies that d|c. On the other hand, c|(a + nb), c|b imply that c|((a + nb) − nb) = a. Thus c is
a common divisor of a and b, implying that c|d. This completes the proof. ❑
329 Example Use Theorem 328 to find (3456, 246).
Solution: (3456, 246) = (13 · 246 + 158, 246) = (158, 246), by the preceding example. Now, (158, 246) = (158, 158 + 88) =
(88, 158). Finally, (88, 158) = (70, 88) = (18, 70) = (16, 18) = (2, 16) = 2. Hence (3456, 246) = 2.
330 Theorem If rn is the last non-zero remainder found in the process of the Euclidean Algorithm, then
rn = (a, b).
48
Euclidean Algorithm
49
Proof: From equations 5.1
r2
r3
r4
..
.
=
=
=
..
.
a − bq1
b − r2 q2
r2 − r3 q3
..
.
rn
=
rn−2 − rn−1 qn−1
Let r = (a, b). From the first equation, r|r2 . From the second equation, r|r3 . Upon iterating the process, we see that
r|rn .
But starting at the last equation 5.1 and working up, we see that rn |rn−1 , rn |rn−2 , . . . rn |r2 , rn |b, rn |a. Thus rn is a
common divisor of a and b and so rn |(a, b). This gives the desired result. ❑
331 Example Find (23, 29) by means of the Euclidean Algorithm.
Solution: We have
29 = 1 · 23 + 6,
23 = 3 · 6 + 5,
6 = 1 · 5 + 1,
5 = 5 · 1.
The last non-zero remainder is 1, thus (23, 29) = 1.
An equation which requires integer solutions is called a diophantine equation. By the Bachet-Bezout Theorem, we see that
the linear diophantine equation
ax + by = c
has a solution in integers if and only if (a, b)|c. The Euclidean Algorithm is an efficient means to find a solution to this equation.
332 Example Find integers x, y that satisfy the linear diophantine equation
23x + 29y = 1.
Solution: We work upwards, starting from the penultimate equality in the preceding problem:
1 = 6 − 1 · 5,
5 = 23 − 3 · 6,
6 = 29 · 1 − 23.
Hence,
1
This solves the equation, with x = −5, y = 4.
=
=
=
=
=
6−1·5
6 − 1 · (23 − 3 · 6)
4 · 6 − 1 · 23
4(29 · 1 − 23) − 1 · 23
4 · 29 − 5 · 23.
333 Example Find integer solutions to
23x + 29y = 7.
Solution: From the preceding example, 23(−5) + 29(4) = 1. Multiplying both sides of this equality by 7,
23(−35) + 29(28) = 7,
which solves the problem.
50
Chapter 5
334 Example Find infinitely many integer solutions to
23x + 29y = 1.
Solution: By Example 332, the pair x0 = −5, y0 = 4 is a solution. We can find a family of solutions by letting
x = −5 + 29t, y = 4 − 23t, t ∈ Z.
335 Example Can you find integers x, y such that 3456x + 246y = 73?
Solution: No. (3456, 246) = 2 and 2 6 |73.
336 Theorem Assume that a, b, c are integers such that (a, b)|c. Then given any solution (x0 , y0 ) of the linear diophantine
equation
ax + by = c
any other solution of this equation will have the form
b
a
x = x0 + t , y = y0 − t ,
d
d
where d = (a, b) and t ∈ Z.
Proof: It is clear that if (x0 , y0 ) is a solution of ax + by = c, then x = x0 + tb/d, y = y0 − ta/d is also a solution.
Let us prove that any solution will have this form.
Let (x′ , y′ ) satisfy ax′ + by′ = c. As ax0 + by0 = c also, we have
a(x′ − x0 ) = b(y0 − y′ ).
Dividing by d = (a, b),
b
a ′
(x − x0 ) = (y0 − y′ ).
d
d
a
a
|(y0 − y′ ), in virtue of Euclid’s Lemma. Thus there is an integer t such that t = y0 − y′ ,
d
d
that is, y = y0 − ta/d. From this
a ′
b a
(x − x0 ) = t ,
d
d d
Since (a/d, b/d) = 1,
which is to say x′ = x0 + tb/d. This finishes the proof. ❑
337 Example Find all solutions in integers to
3456x + 246y = 234.
Solution: By inspection, 3456(−1) + 246(15) = 234. By Theorem 336, all the solutions are given by x = −1 + 123t, y =
15 − 1728t,t ∈ Z.
Practice
338 Problem Find the following:
4. (8098643070, 8173826342)
1. (34567, 987)
2. (560, 600)
3. (4554, 36)
339 Problem Solve the following linear diophantine equations, provided solutions exist:
Linear Congruences
51
1. 24x + 25y = 18
tices are (0, 0), (b, a), (x, y) is
|by − ax|
.
2
2. 3456x + 246y = 44
3. 1998x + 2000y = 33
340 Problem Prove that the area of the triangle whose ver-
341 Problem A woman pays $2.78 for some bananas and
eggs. If each banana costs $0.69 and each egg costs $0.35,
how many eggs and how many bananas did the woman buy?
5.2 Linear Congruences
We recall that the expression ax ≡ b mod n means that there is t ∈ Z such that ax = b + nt. Hence, the congruencial equation
in x, ax ≡ b mod n is soluble if and only if the linear diophantine equation ax + ny = b is soluble. It is clear then that the
congruence
ax ≡ b mod n
has a solution if and only if (a, n)|b.
342 Theorem Let a, b, n be integers. If the congruence ax ≡ b mod n has a solution, then it has (a, n) incongruent solutions
mod n.
Proof: From Theorem 336 we know that the solutions of the linear diophantine equation ax + ny = b have the
form x = x0 + nt/d, y = y0 − at/d, d = (a, n),t ∈ Z, where x0 , y0 satisfy ax0 + ny = b. Letting t take on the values
t = 0, 1, . . . ((a, n) − 1), we obtain (a, n) mutually incongruent solutions, since the absolute difference between any
two of them is less than n. If x = x0 + nt ′ /d is any other solution, we write t ′ as t ′ = qd + r, 0 ≤ r < d. Then
x
= x0 + n(qd + r)/d
= x0 + nq + nr/d
≡ x0 + nr/d mod n.
Thus every solution of the congruence ax ≡ b mod n is congruent mod n to one and only one of the d values
x0 + nt/d, 0 ≤ t ≤ d − 1. Thus if there is a solution to the congruence, then there are d incongruent solutions
mod n.❑
343 Example Find all solutions to the congruence 5x ≡ 3 mod 7
Solution: Notice that according to Theorem 342, there should only be one solution mod 7, as (5, 7) = 1. We first solve the
linear diophantine equation 5x + 7y = 1. By the Euclidean Algorithm
=
=
=
5·1+2
2·2+1
2 · 1.
1 =
2 =
5−2·2
7 − 5 · 1,
7
5
2
Hence,
which gives
1 = 5 − 2 · 2 = 5 − 2(7 − 5 · 1) = 5 · 3 − 7 · 2.
Whence 3 = 5(9) − 7(6). This gives 5 · 9 ≡ 3 mod 7 which is the same as 5 · 2 ≡ 3 mod 7. Thus x ≡ 2 mod 7.
344 Example Solve the congruence
3x ≡ 6
mod 12.
52
Chapter 5
Solution: As (3, 12) = 3 and 3|6, the congruence has three mutually incongruent solutions. By inspection we see that x = 2 is
a solution. By Theorem 336, all the solutions are thus of the form x = 2 + 4t,t ∈ Z. By letting t = 0, 1, 2, the three incongruent
solutions modulo 12 are t = 2, 6, 10.
We now add a few theorems and definitions that will be of use in the future.
345 Theorem Let x, y be integers and let a, n be non-zero integers. Then
ax ≡ ay
mod n
if and only if
x≡y
mod
n
.
(a, n)
Proof: If ax ≡ ay mod n then a(x − y) = sn for some integer s. This yields
(x − y)
a
n
=s
.
(a, n)
(a, n)
Since (a/(a, n), n/(a, n)) = 1 by Theorem 240, we must have
n
|(x − y),
(a, n)
by Euclid’s Lemma (Lemma 239). This implies that
x≡y
Conversely if x ≡ y mod
mod
n
.
(a, n)
n
implies
(a, n)
ax ≡ ay
mod
an
,
(a, n)
upon multiplying by a. As (a, n) divides a, the above congruence implies a fortiori that ax − ay = tn for some
integer t. This gives the required result.❑
Theorem 345 gives immediately the following corollary.
346 Corollary If ax ≡ ay mod n and (a, n) = 1, then x ≡ y mod n.
Practice
347 Problem Solve the congruence 50x ≡ 12 mod 14.
348 Problem How many x, 38 ≤ x ≤ 289 satisfy
3x ≡ 8
mod 11?
5.3 A theorem of Frobenius
If (a, b) = d > 1 then the linear form ax + by skips all non-multiples of d. If (a, b) = 1, there is always an integer solution
to ax + by = n regardless of the integer n. We will prove the following theorem of Frobenius that tells un when we will find
nonnegative solutions to ax + by = n.
349 Theorem (Frobenius) Let a, b be positive integers. If (a, b) = 1 then the number of positive integers m that cannot be
written in the form ar + bs = m for nonnegative integers r, s equals (a − 1)(b − 1)/2.
A theorem of Frobenius
53
Proof: Let us say that an integer n is attainable if there are nonnegative integers r, s with ar + bs = n. Consider
the infinite array
0
1
2 ...
k ... a−1
a
a+1
a + 2 . . . a + k . . . 2a − 1
2a 2a + 1 2a + 2 . . . 2a + k . . . 3a − 1
...
...
... ...
... ...
...
The columns of this array are arithmetic progressions with common difference a. The numbers directly below a
number n have the form n + ka where k is a natural number. Clearly, if n is attainable, so is n + ka, implying thus
that if an integer n is attainable so is every integer directly below it. Clearly all multiples of b are attainable. We
claim that no two distinct multiples of b, vb and wb with 0 ≤ v, w ≤ a − 1 can belong to the same column. If this
were so then we would have vb ≡ wb mod a. Hence a(v − w) ≡ 0 mod a. Since (a, b) = 1 we invoke Corollary
5.1 to deduce v − w ≡ 0 mod a. Since 0 ≤ v, w ≤ a − 1, we must have v = w.
Now we show that any number directly above one of the multiples vb, 0 ≤ v ≤ a − 1 is non-attainable. For a number
directly above vb is of the form vb−ka for some natural number k. If vb−ka were attainable, then ax+by = vb−ka
for some nonnegative integers x, y. This yields by ≤ ax + by = vb − ka < vb. Hence, 0 ≤ y < v < a. This implies
that y 6≡ v mod b. On the other hand, two numbers on the same column are congruent mod a. Therefore we
deduce vb ≡ bv − ka ≡ ax + by mod a which yields bv ≡ by mod a. By Corollary 346 we obtain v ≡ y mod a.
This contradicts the fact that 0 ≤ y < v < a.
Thus the number of unattainable numbers is precisely the numbers that occur just above a number of the form
vb, 0 ≤ v ≤ a − 1. Now, on the j-th column, there are (vb − j)/a values above vb. Hence the number of unattainable
numbers is given by
a−1 X
a−1
X
v=0 j=0
vb − j (a − 1)(b − 1)
=
,
a
2
as we wanted to show.❑
The greatest unattainable integer occurs just above (a − 1)b, hence the greatest value that is not attainable is (a − 1)b − a,
which gives the following theorem.
350 Theorem Let a, b be relatively prime positive integers. Then the equation
ax + by = n
is unsoluble in nonnegative integers x, y for n = ab−a−b. If n > ab−a−b, then the equation is soluble in nonnegative integers.
351 Example (Putnam, 1971) A game of solitaire is played as follows. After each play, according to the outcome, the player
receives either a or b points, (a, b ∈ N, a > b), and his score accumulates from play to play. It has been noticed that there are
thirty five non-attainable scores and that one of these is 58. Find a and b.
Solution: The attainable scores are the nonnegative integers of the form ax + by. If (a, b) > 1, there are infinitely many such
integers. Hence (a, b) = 1. By Theorem 349, the number of non-attainable scores is (a−1)(b−1)/2. Therefore, (a−1)(b−1) =
70 = 2(35) = 5(14) = 7(10). The conditions a > b, (a, b) = 1 yield the two possibilities a = 71, b = 2 and a = 11, b = 8. As
58 = 0 · 71 + 2 · 29, the first alternative is dismissed. The line 11x + 8y = 58 passes through (6, −1) and (−2, 10) and thus it
does not pass through a lattice point in the first quadrant. The unique solution is a = 11, b = 8.
352 Example (AIME, 1994) Ninety-four bricks, each measuring 4′′ × 10′′ × 19′′ , are to be stacked one on top of another to
form a tower 94 bricks tall. Each brick can be oriented so it contributes 4′′ or 10′′ or 19′′ to the total height of the tower. How
many different tower heights can be achieved using all 94 of the bricks?
54
Chapter 5
Solution: Let there be x, y, z bricks of height 4′′ , 10′′ , and 19′′ respectively. We are asking for the number of different sums
4x + 10y + 19z
with the constraints x ≥ 0, y ≥ 0, z ≥ 0, x + y + z = 94.
Now, 4x + 10y + 19z ≤ 19 · 94 = 1786. Letting x = 94 − y − z, we count the number of different nonnegative integral
solutions to the inequality 376 + 3(2y + 5z) ≤ 1786, y + z ≤ 94, that is 2y + 5z ≤ 470, y + z ≤ 94. By Theorem 350, every integer
≥ (2 − 1)(5 − 1) = 4 can be written in the form 2y + 5z, and the number of exceptions is (2 − 1)(5 − 1)/2 = 2, namely n = 1 and
n = 3. Thus of the 471 nonnegative integers n ≤ 470, we see that 469 can be written in the form n = 2y+5z. Using x = 96−x−y,
n, 4 ≤ n ≤ 470 will be “good” only if we have 470 − n = 3x + 5z. By Theorem 349 there are (3 − 1)(5 − 1)/2 = 4 exceptions,
each ≤ 8, namely n = 1, 2, 4, 7. This means that 463, 466, 468, and 469 are not representable in the form 4x + 10y + 19z. Then
every integer n, 0 ≤ n ≤ 470 except for 1, 3, 463, 466, 468, and 469 can be thus represented, and the number of different sums
is 471 − 6 = 465.
n
is the sum of two positive integers with denominator < 1991 if an
1. Let (n, 1991) = 1. Prove that
1991
only if there exist integers m, a, b with
353 Example
(∗)
1 ≤ m ≤ 10, a ≥ 1, b ≥ 1, mn = 11a + 181b.
2. Find the largest positive rational with denominator 1991 that cannot be written as the sum of two positive rationals each
with denominators less than 1991.
n
a
b
n
a b
=
+
does the trick. Conversely, if
= + for a, b ≥ 1, (a, r) = (b, s) = 1,
1991 181m 11m
1991 r s
and r, s < 1991, we may suppose r = 181r1 , s = 11s1 and then nr1 s1 = 11as1 + 181br1 , which leads to r1 |11as1 and so r1 |s1 .
Similarly, s1 |r1 , whence r1 = s1 = m, say, and (∗) follows.
(b) Any n > 170, (n, 1991) = 1 satisfies (∗) with b = 1 and M such that mn is of the form mn ≡ 181 mod 11. For mn > 181
except if m = 1, n ≤ 180; but then n would not be of the form n ≡ 181 mod 11.
Solution: (a) If (∗) holds then
But n = 170 does not satisfy (∗); for we would have 170 ≡ 181b mod 11, so b ≡ m mod 11, which yields b ≥ m, but
170m < 181. The answer is thus 170/1991.
Practice
354 Problem Let a, b, c be positive real numbers. Prove that
there are at least c2 /2ab pairs of integers (x, y) satisfying
357 Problem Let a, b ∈ N, (a, b) = 1. Let S(n) denote the
number of nonnegative solutions to
x ≥ 0, y ≥ 0, ax + by ≤ c.
ax + by = n.
355 Problem (AIME, 1995) What is largest positive integer
that is not the sum of a positive integral multiple of 42 and
a positive composite integer?
356 Problem Let a > 0, b > 0, (a, b) = 1. Then the number of
nonnegative solutions to the equation ax + by = n is equal to
n
n
[ ] or [ ] + 1.
ab
ab
(Hint: [s] − [t] = [s − t] or [s − t] + 1.)
Evaluate
lim
n→∞
S(n)
.
n
358 Problem (IMO, 1983) Let a, b, c be pairwise relatively
prime integers. Demonstrate that 2abc − ab − bc − ca is the
largest integer not of the form
bcx + acy + abz,
x ≥ 0, y ≥ 0, z ≥ 0.
Chinese Remainder Theorem
55
5.4 Chinese Remainder Theorem
In this section we consider the case when we have multiple congruences. Consider the following problem: find an integer x
which leaves remainder 2 when divided by 5, is divisible by 7, and leaves remainder 4 when divided by 11. In the language of
congruences we are seeking x such that
x ≡ 2
mod 5,
x ≡ 0
mod 7,
x ≡ 4
mod 11.
One may check that x = 147 satisfies the requirements, and that in fact, so does the parametric family x = 147 + 385t,t ∈ Z.
We will develop a method to solve congruences like this one. The method is credited to the ancient Chinese, and it is thus
called the Chinese Remainder Theorem.
359 Example Find x such that
x≡3
mod 5 and x ≡ 7
mod 11.
Solution: Since x = 3 + 5a, we have 11x = 33 + 55a. As x = 7 + 11b, we have 5x = 35 + 55b. Thus x = 11x − 10x = 33 − 70 +
55a − 110b. This means that x ≡ −37 ≡ 18 mod 55. One verifies that all the numbers x = 18 + 55t,t ∈ Z verify the given
congruences.
360 Example Find a number n such that when divided by 4 leaves remainder 2, when divided by 5 leaves remainder 1, and
when divided by 7 leaves remainder 1.
Solution: We want n such that
n≡ 2
n≡ 1
n≡ 1
mod 4,
mod 5,
mod 7.
This implies that
35n ≡ 70
28n ≡ 28
20n ≡ 20
do.
mod 140,
mod 140,
mod 140.
As n = 21n − 20n, we have n ≡ 3(35n − 28n) − 20n ≡ 3(70 − 28) − 20 ≡ 106 mod 140. Thus all n ≡ 106 mod 140 will
361 Theorem (Chinese Remainder Theorem) Let m1 , m2 , . . . mk be pairwise relatively prime positive integers, each exceeding 1, and let a1 , a2 , . . . ak be arbitrary integers. Then the system of congruences
x
x
..
.
≡
≡
..
.
a1
a2
..
.
mod m1
mod m2
x
≡ ak
mod mk
has a unique solution modulo m1 m2 · · · mk .
Proof: Set Pj = m1 m2 · · · mk /m j , 1 ≤ j ≤ k. Let Q j be the inverse of Pj mod m j , i.e., Pj Q j ≡ 1 mod m j , which
we know exists since all the mi are pairwise relatively prime. Form the number
x = a1 P1 Q1 + a2 P2 Q2 + · · · + ak Pk Qk .
This number clearly satisfies the conditions of the theorem. The uniqueness of the solution modulo m1 m2 · · · mk can
be easily established. ❑
56
Chapter 5
362 Example Can one find one million consecutive integers that are not square-free?
Solution: Yes. Let p1 , p2 , . . . , p1000000 be a million different primes. By the Chinese Remainder Theorem, there exists a solution
to the following system of congruences.
x
x
..
.
≡
≡
..
.
x
≡ −1000000
−1
−2
..
.
..
.
mod p21 ,
mod p22 ,
mod p21000000 .
The numbers x + 1, x + 2, . . . , x + 1000000 are a million consecutive integers, each of which is divisible by the square of a prime.
Practice
363 Problem Solve the following systems:
1. x ≡ −1 mod 4; x ≡ 2 mod 5
2. 4x ≡ 3 mod 7; x ≡ 10 mod 11
3. 5x ≡ 2 mod 8; 3x ≡ 2 mod 9; x ≡ 0 mod 11
1. Do there exist fourteen
consecutive positive integers each of which is divisible
by one or more primes p, 2 ≤ p ≤ 11?
364 Problem (USAMO 1986)
2. Do there exist twenty-one consecutive integers each of
which is divisible by one or more primes p, 2 ≤ p ≤ 13?
6
Chapter
Number-Theoretic Functions
6.1 Greatest Integer Function
The largest integer not exceeding x is denoted by TxU or TxU. We also call this function the floor function. Thus TxU satisfies
the inequalities x − 1 < TxU ≤ x, which, of course, can also be written as TxU ≤ x < TxU + 1. The fact that TxU is the unique
integer satisfying these inequalities, is often of use. We also utilise the notation {x} = x − TxU, to denote the fractional part of
x, and ||x|| = min |x − n| to denote the distance of a real number to its nearest integer. A useful fact is that we can write any real
n∈Z
number x in the form x = TxU + {x}, 0 ≤ {x} < 1.
The greatest integer function enjoys the following properties:
365 Theorem Let α , β ∈ R, a ∈ Z, n ∈ N. Then
1. Tα + aU = Tα U + a
α
Tα U
2. T U = T
U
n
n
3. Tα U + Tβ U ≤ Tα + β U ≤ Tα U + Tβ U + 1
Proof:
1. Let m = Tα + aU. Then m ≤ α + a < m + 1. Hence m − a ≤ α < m − a + 1. This means that m − a = Tα U,
which is what we wanted.
2. Write α /n as α /n = Tα /nU + θ , 0 ≤ θ < 1. Since nTα /nU is an integer, we deduce by (1) that
Tα U = TnTα /nU + nθ U = nTα /nU + Tnθ U.
Now, 0 ≤ Tnθ U ≤ nθ < n, and so 0 ≤ Tnθ U/n < 1. If we let Θ = Tnθ U/n, we obtain
α
Tα U
= T U + Θ, 0 ≤ Θ < 1.
n
n
This yields the required result.
3. From the inequalities α − 1 < Tα U ≤ α , β − 1 < Tβ U ≤ β we get α + β − 2 < Tα U + Tβ U ≤ α + β . Since
Tα U + Tβ U is an integer less than or equal to α + β , it must be less than or equal to the integral part of α + β ,
i.e. Tα + β U. We obtain thus Tα U + Tβ U ≤ Tα + β U. Also, α + β is less than the integer Tα U + Tβ U + 2, so
its integer part Tα + β U must be less than Tα U + Tβ U + 2, but Tα + β U < Tα U + Tβ U + 2 yields Tα + β U ≤
Tα U + Tβ U + 1. This proves the inequalities.
❑
57
58
Chapter 6
366 Example Find a non-zero polynomial P(x, y) such that
P(T2tU, T3tU) = 0
for all real t.
Solution: We claim that 3[2t] − 2[3t] = 0, ±1 or −2. We can then take
P(x, y) = (3x − 2y)(3x − 2y − 1)(3x − 2y + 1)(3x − 2y + 2).
In order to prove the claim, we observe that TxU has unit period, so it is enough to prove the claim for t ∈ [0, 1). We divide
[0, 1) as
[0, 1) = [0, 1/3) ∪ [1/3, 1/2) ∪ [1/2, 2/3) ∪ [2/3, 1).
If t ∈ [0, 1/3), then both T2tU and T3tU are = 0, and so 3T2tU − 2T3tU = 0. If t ∈ [1/3, 1/2) then [3t] = 1 and [2t] = 0, and so
3T2tU − 2T3tU = −2. If t ∈ [1/2, 2/3), then [2t] = 1, [3t] = 1, and so 3T2tU − 2T3tU = 1. If t ∈ [2/3, 1), then T2tU = 1, T3tU = 2,
and 3T2tU − 2T3tU = −1.
√
367 Example Describe all integers n such that 1 + T 2nU 2n.
√
√
√
√
√
Solution: Let 2n = m(1√+ T 2nU). If m ≤ T √2nU − 1 then 2n ≤ (T 2nU − 1)(T 2nU + 1) = T 2nU2 − 1 ≤ 2n − 1 <√2n, a
contradiction. If m ≥ T 2nU+1, then 2n ≥ (T 2nU2 +1)2 ≥ 2n+1, another contradiction. It must be the case that m = T 2nU.
Conversely, let n =
triangular numbers.
√
√
l(l + 1)
. Since l < 2n < l + 1, l = T 2nU. So all the integers with the required property are the
2
368 Example Prove that the integers
€
√ Šn
T 1+ 2 U
with n a nonnegative integer, are alternately even or odd.
Solution: By the Binomial Theorem
√
√
(1 + 2)n + (1 − 2)n = 2
X
(2)k
0≤k≤n/2
n
:= 2N,
2k
√
√
√
√
of (1 + 2)√n or (1 + 2)√n + 1
an even integer. Since −1 < 1 − 2 < 0, it must be the case that (1 − 2)n is√the fractional part
√
depending on√whether n is√odd or even,√respectively. Thus for odd n, (1 + 2)n − 1 <√(1 + 2)n +√(1 − 2)n < √
(1 + 2)n ,
n
n
n
n
n
whence (1 + √2) + (1 − 2) = T(1 + 2) U, always even, and for n even 2N := (1 + 2) + (1 − 2) = T(1 + 2)n U + 1,
and so T(1 + 2)n U = 2N − 1, always odd for even n.
369 Example Prove that the first thousand digits after the decimal point in
√
(6 + 35)1980
are all 9’s.
Solution: Reasoning as in the preceding problem,
√
√
(6 + 35)1980 + (6 − 35)1980 = 2k,
Greatest Integer Function
59
√
√
1
< 6 − 35, upon squaring 3500 < 3481, which is clearly nonsense), and
an even integer. But 0 < 6 − 35 < 1/10, (for if
10
√
hence 0 < (6 − 35)1980 < 10−1980 which yields
√
1
2k − 1 + |0.9{z
. . . 9} = 2k − 1980 < (6 + 35)1980 < 2k,
10
1979 nines
This proves the assertion of the problem.
370 Example (Putnam 1948) If n is a positive integer, demonstrate that
√
√
√
T n + n + 1U = T 4n + 2U.
Solution: By squaring, it is easy to see that
√
√
√
√
4n + 1 < n + n + 1 < 4n + 3.
Neither 4n + 2 nor 4n + 3 are squares since squares are either congruent to 0 or 1 mod 4, so
√
√
T 4n + 2U = T 4n + 3U,
and the result follows.
371 Example Find a formula for the n-th non-square.
Solution: Let Tn be the n-th non-square. There is a natural number m such that m2 < Tn < (m + 1)2 . As there are m squares less
than Tn and n non-squares up to Tn , we see that Tn = n + m. We have then m2 < n + m < (m + 1)2 or m2 − m < n < m2 + m + 1.
1
1
Since n, m2 − m, m2 + m + 1 are all integers, these inequalities imply m2 − m + < n < m2 + m + , that is to say, (m − 1/2)2 <
4
4
√
√
1
2
n < (m + 1/2) . But then m = T n + U. Thus the n-th non-square is Tn = n + T n + 1/2U.
2
√
372 Example (Putnam 1983) Let f (n) = n + T nU. Prove that for every positive integer m, the sequence
m, f (m), f ( f (m)), f ( f ( f (m))), . . .
contains at least one square of an integer.
Solution: Let m = k2 + j, 0 ≤ j ≤ 2k. Split the m’s into two sets, the set A of all the m with excess j, 0 ≤ j ≤ k and the set B
with all those m’s with excess j, k < j < 2k + 1.
√
Observe that k2 ≤ m < (k + 1)2 = k2 + 2k + 1. If j = 0, we have nothing to prove. Assume that m ∈ B. As T mU = k,
f (m) = k2 + j + k = (k + 1)2 + j − k − 1, with 0 ≤ j − k − 1 ≤ k − 1 <
√k + 1. This means that either f (m) is a square or f (m) ∈ A.
It is thus enough to consider the alternative m ∈ A, in which case T m + kU = k and
f ( f (m)) = f (m + k) = m + 2k = (k + 1)2 + j − 1.
This means that f ( f (m)) is either a square or f ( f (m)) ∈ A with an excess j − 1 smaller than the excess j of m. At each iteration
the excess will reduce and eventually it will hit 0, whence we reach a square.
373 Example Solve the equation
Tx2 − x − 2U = TxU,
for x ∈ R.
Solution: Observe that TaU = TbU if and only if ∃k ∈ Z with a, b ∈ [k, k + 1) which happens if and only if |a − b| < 1. Hence,
the given equation has a solution if and only if |x2 − 2x − 2| < 1. Solving these inequalities it is easy to see that the solution is
thus
√
√
√
1
1
1
x ∈ (−1, (1 − 5)] ∪ [ (1 + 17), (1 + 21)).
2
2
2
60
Chapter 6
374 Theorem If a, b are relatively prime natural numbers then
a−1
X
b−1
k=1
k=1
X ka
(a − 1)(b − 1)
kb
.
T U=
T U=
a
b
2
Consider the rectangle with vertices at (0, 0), (0, b), (a, 0), (a, b). This rectangle contains (a − 1)(b − 1)
xb
lattice points, i.e., points with integer coordinates. This rectangle is split into two halves by the line y = .
a
We claim that there are no lattice points on this line, except for the endpoints. For if there were a lattice point
n b
(m, n), 0 < m < a, 0 < n < b, then = . Thus n/m is a reduction for the irreducible fraction b/a, a contradiction.
m a
kb
kb
The points Lk = (k, ), 1 ≤ k ≤ a − 1 are each on this line. Now, T U equals the number of lattice points on the
a
a
a−1
X
kb
kb
vertical line that goes from (k, 0) to (k, ), i.e.
T U is the number of lattice points on the lower half of the
a
a
Proof:
k=1
b−1
X
ka
rectangle. Similarly,
T U equals the number of lattice points on the upper half of the rectangle. Since there
b
k=1
are (a − 1)(b − 1) lattice points in total, and their number is shared equally by the halves, the assertion follows. ❑
375 Example Find the integral part of
6
10
X
1
√ .
k
k=1
Solution: The function x 7→ x−1/2 is decreasing. Thus for positive integer k,
Z k+1
1
√
<
k+1
k
dx
1
√ <√ .
x
k
6
Summing from k = 1 to k = 10 − 1 we deduce
6
10
X
1
√ <
k
k=2
Z 106
1
dx
√ <
x
6 −1
10
X
k=1
1
√ .
k
The integral is easily seen to be 1998. Hence
6
3
1998 + 1/10 <
10
X
1
√ < 1999.
k
k=1
The integral part sought is thus 1998.
Practice
376 Problem Prove that for all real numbers x, y,
378 Problem If n > 1 is a natural number and α ≥ 1 is a real
number, prove that
α
[α ] > T U.
n
TxU + Tx + yU + TyU ≤ T2xU + T2yU
holds.
379 Problem If a, b, n are positive integers, prove that
377 Problem If x, y real numbers, when is it true that
TxUTyU ≤ TxyU?
T
ab
b
U ≥ aT U.
n
n
Practice
61
380 Problem Let α be a real number. Prove that [α ]+[−α ] =
−1 or 0 and that Tα U − 2Tα /2U = 0 or 1.
390 Problem (Putnam 1973) Prove that if n ∈ N,
√
min(k + Tn/kU) = T 4n + 1U.
k∈N
381 Problem Prove that
√
T(2 + 3)n U
391 Problem (Dirichlet’s principle of the hyperbola) Let
N be the number of integer solutions to xy ≤ n, x > 0, y > 0.
Prove that
is an odd integer.
N=
382 Problem Show that the n-th element of the sequence
n
X
X
k=1
√
1≤k≤ n
n
T U=2
k
√
n
T U − T nU2 .
k
1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, . . .
392 Problem (Circle Problem) Let r > 0 and let T denote the
√
where there are n occurrences of the integer n is T 2n + 1/2U.
number of lattice points of the domain x2 + y2 ≤ r2 . Prove that
X
p
r
T r2 − x2 U + 4T √ U2 .
T = 1 + 4TrU + 8
√
2
383 Problem Prove Hermite’s Identity: if x is a real number
and n is a natural number then
1
2
n−1
TnxU = TxU + Tx + U + Tx + U + · · · + Tx +
U.
n
n
n
0<x≤r 2
393 Problem Let d = (a, b). Prove that
X
1≤n≤b−1
384 Problem Prove that for all integers m, n, the equality
T
m+n
n−m+1
U+T
U=n
2
2
holds.
TnaU + TnbU = TncU + TndU
for all natural numbers n, prove that
a + b = c + d.
(a − 1)(b − 1) d − 1
an
U=
+
.
b
2
2
394 Problem (Eisenstein) If (a, b) = 1 and a, b are odd, then
X
1≤n≤(b−1)/2
385 Problem If a, b, c, d are positive real numbers such that
T
T
an
U+
b
X
1≤n≤(a−1)/2
T
(a − 1)(b − 1)
bn
U=
.
a
4
395 Problem Let m ∈ N with m > 1 and let y be a positive
real number. Prove that
X
É
Tm
x
y
U = TyU,
x
where the summation runs through all positive integers x not
divisible by the mth power of an integer exceeding 1.
386 Problem If n is a natural number, prove that
396 Problem For which natural numbers n will 112 divide
n + 2 − Tn/25U
8n + 24
T
U=T
U.
3
25
387 Problem Solve the equation
T
x
x
U=T
U.
1994
1995
388 Problem Let [α , β ] be an interval which contains no inte-
√
4n − T(2 + 2)n U?
397 Problem A triangular number is a number of the form
1 + 2 + · · · + n, n ∈ N. Find a formula for the nth non-triangular
number.
398 Problem (AIME 1985) How many of the first thousand
gers. Prove that there is a positive integer n such that [nα , nβ ]
still contains no integers but has length at least 1/6.
positive integers can be expressed in the form
389 Problem (IMO 1968) For every natural number n, evalu-
399 Problem (AIME 1987) What is the largest positive integer n for which there is a unique integer k such that
ate the sum
∞
X
T
k=0
n + 2k
U.
2k+1
T2xU + T4xU + T6xU + T8xU?
8
n
7
<
< ?
15 n + k 13
62
Chapter 6
Find the value of T100rU.
400 Problem Prove that if p is an odd prime, then
√
T(2 + 5) p U − 2 p+1
406 Problem (AIME 1995) Let f (n) denote the integer clos-
is divisible by p.
est to n1/4 , when n is a natural number. Find the exact numerical value of
1995
X 1
401 Problem Prove that the n-th number not of the form
k
.
Te U, k = 1, 2, . . . is
f (n)
n=1
Tn = n + Tln(n + 1 + Tln(n + 1)U)U.
407 Problem Prove that
402 Problem (Leningrad Olympiad) How many different
integers are there in the sequence
T
12
1980
U, T
22
1980
U, . . . , T
Z 1
T1994xU+T1995xU
(−1)
19802
1980
0
1993
T1994xU
1994
dx = 0.
T1995xU
U?
408 Problem Prove that
√
√
√
√
T n + n + 1U = T n + n + 2U.
403 Problem Let k ≥ 2 be a natural number and x a positive
real number. Prove that
È
√
T k xU = T k TxUU.
409 Problem (Putnam 1976) Prove that
2. Prove that for any real number x 6= 0 at least one of
x, 2x, . . . 79x has a 7 in its decimal expansion.

X
1. Find a real number x 6= 0 such that
x, 2x, . . . , 34x have no 7’s in their decimal expansions.
n→∞
‹
T
lim
404 Problem
1≤k≤n
n
2n
U − 2T U = ln 4 − 1.
k
k
410 Problem (Putnam 1983) Prove that
1
lim
n→∞ n
3. Can you improve the “gap” between 34 and 79?
Z n
1
n
x
dx = log3 (4/π ).
405 Problem (AIME 1991) Suppose that r is a real number
for which
You may appeal to Wallis Product Formula:
91
X
Tr +
k=19
k
U = 546.
100
π
2 2 4 4 6 6 8 8
· · · · · · · ··· = .
1 3 3 5 5 7 7 9
2
6.2 De Polignac’s Formula
We will consider now the following result due to De Polignac.
411 Theorem (De Polignac’s Formula) The highest power of a prime p dividing n! is given by
∞
X
T
k=1
n
U.
pk
Proof: The number of integers contributing a factor of p is Tn/pU, the number of factors contributing a second
factor of p is Tn/p2 U, etc..❑
412 Example How many zeroes are at the end of 300!?
Solution: The number of zeroes is determined by how many times 10 divides into 300. Since there are more factors of 2 in 300!
than factors of 5, the number of zeroes is thus determined by the highest power of 5 in 300!. By De Polignac’s Formula this is
∞
X
T300/5k U = 60 + 12 + 2 = 74.
k=1
De Polignac’s Formula
63
413 Example Does
7
1000
?
500
2
Solution: The highest power of 7 dividing into 1000! is T1000/7U + T1000/7
U + T1000/73 U = 142 + 20 + 2 = 164. Similarly,
1000
1000!
, the highest power of 7 that divides
the highest power of 7 dividing into 500! is 71 + 10 + 1 = 82. Since
=
(500!)2
500
1000
1000
is 164 − 2 · 82 = 0, and so 7 does not divide
.
500
500
414 Example Let n = n1 + n2 + · · · + nk where the ni are nonnegative integers. Prove that the quantity
n!
n1 !n2 ! · · · nk !
is an integer.
Solution: From (3) in Theorem 365 we deduce by induction that
Ta1 U + Ta2 U + · · · + Tal U ≤ Ta1 + a2 + · · · + al U.
For any prime p, the power of p dividing n! is
X
X
j≥1
j≥1
Tn/p j U =
The power of p dividing n1 !n2 ! · · · nk ! is
T(n1 + n2 + · · · + nk )/p j U.
X
j≥1
Tn1 /p j U + Tn2 /p j U + · · · Tnk /p j U.
Since
Tn1 /p j U + Tn2 /p j U + · · · + Tnk /p j U ≤ T(n1 + n2 + · · · + nk )/p j U,
we see that the power of any prime dividing the numerator of
n!
n1 !n2 ! · · · nk !
is at least the power of the same prime dividing the denominator, which establishes the assertion.
415 Example Given a positive integer n > 3, prove that the least common multiple of the products x1 x2 · · · xk (k ≥ 1), whose
factors xi are the positive integers with
x1 + x2 + · · · xk ≤ n,
is less than n!.
Solution: We claim that the least common multiple of the numbers in question is
Y
pTn/pU .
p
p prime
Consider an arbitrary product x1 x2 · · · xk , and an arbitrary prime p. Suppose that pα j |x j , pα j +1 6 |x j . Clearly pα1 + · · · + pαk ≤ n
and since pα ≥ α p, we have
n
p(α1 + · · · αk ) ≤ n or α1 + · · · + αk ≤ T U.
p
Hence it follows that the exponent of an arbitrary prime p is at most Tn/pU. But on choosing x1 = · · · = xk = p, k = Tn/pU, we
see that there is at least one product for which equality is achieved. This proves the claim.
The assertion of the problem now follows upon applying De Polignac’s Formula and the claim.
64
Chapter 6
Practice
416 Problem (AHSME 1977) Find the largest possible n such
is an integer.
that 10n divides 1005!.
417 Problem Find the highest power of 17 that divides (17n −
2)! for a positive integer n.
423 Problem (AIME 1992) Define a positive integer n to be a
“factorial tail” if there is some positive integer m such that the
base-ten representation of m! ends with exactly n zeroes. How
many positive integers less than 1992 are not factorial tails?
418 Problem Find the exponent of the highest power of 24
that divides 300!.
424 Problem Prove that if m and n are relatively prime posi-
419 Problem Find the largest power of 7 in 300!.
tive integers then
(m + n − 1)!
m!n!
420 Problem (AIME 1983) What is the largest two-digit
is an integer.
prime factor of the integer
200
?
100
421 Problem (USAMO 1975)
425 Problem If p is a prime divisor of
1. Prove that
T5xU + T5yU ≥ T3x + yU + T3y + xU.
2. Using the result of part 1 or otherwise, prove that
(5m)!(5n)!
m!n!(3m + n)!(3n + m)!
2n
n
with p ≥
prove that the exponent of p in the factorisation of
426 Problem Prove that
 lcm
‹
n
n
n
,
,...,
1
2
n
=
lcm(1, 2, . . . , n + 1)
.
n+1
427 Problem Prove the following result of Catalan:
(2n − 4)!
n!(n − 2)!
2n
n
equals 1.
is an integer for all positive integers m, n.
422 Problem Prove that if n > 1, (n, 6) = 1, then
√
2n
2m
divides
m
m+n
n
2n
.
n
6.3 Complementary Sequences
We define the spectrum of a real number α to be the infinite multiset of integers
Spec(α ) = {Tα U, T2α U, T3α U, . . .}.
Two sequences Spec(α ) and Spec(β ) are said to be complementary if they partition the natural numbers, i.e. Spec(α ) ∩
Spec(β ) = ∅ and Spec(α ) ∪ Spec(β ) = N.
For example, it appears that the two sequences
√
Spec( 2) = {1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, 25, . . .},
and
√
Spec(2 + 2) = {3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, 47, 51, . . .}
are complementary. The following theorem establishes a criterion for spectra to be complementary.
Practice
65
428 Theorem (Beatty’s Theorem, 1926) If α > 1 is irrational and
1 1
+ = 1,
α β
then the sequences
Spec(α ) and Spec(β )
are complementary.
Proof: Since α > 1, β > 1, Spec(α ) and Spec(β ) are each sequences of distinct terms, and the total number of
terms not exceeding N taken together in both sequences is TN/α U + TN/β U. But N/α − 1 + N/β − 1 < TN/α U +
[N/β ] < N/α +N/β , the last inequality being strict because both α , β are irrational. As 1/α +1/β = 1, we gather
that N −2 < TN/α U+TN/β U < N. Since the sandwiched quantity is an integer, we deduce [N/α ]+[N/β ] = N −1.
Thus the total number of terms not exceeding N in Spec(α ) and Spec(β ) is N − 1, as this is true for any N ≥ 1 each
interval (n, n + 1) contains exactly one such term. It follows that Spec(α ) ∪ Spec(β ) = N, Spec(α ) ∩ Spec(β ) = ∅.
❑
The converse of Beatty’s Theorem is also true.
429 Theorem (Bang’s Theorem, 1957) If the sequences
Spec(α ) and Spec(β )
are complementary, then α , β are positive irrational numbers with
1 1
+ = 1.
α β
Proof: If both α , β are rational numbers, it is clear that Spec(α ), Spec(β ) eventually contain the same integers,
and so are not disjoint. Thus α and β must be irrational. If 0 < α ≤ 1, given n there is an M for which
mα − 1 < n ≤ mα ; hence n = [mα ], which implies that Spec(α ) = N, whence α > 1 (and so β > 1 also). If
Spec(α ) ∩ Spec(β ) is finite, then
Tn/α U + Tn/β U
= 1,
lim
n→∞
n
1
but since (Tn/α U + Tn/β U) → 1/α + 1/β as n → ∞, it follows that 1/α + 1/β = 1. ❑
n
430 Example Suppose we sieve the positive integers as follows: we choose a1 = 1 and then delete a1 + 1 = 2. The next term
is 3, which we call a2 , and then we delete a2 + 2 = 5. Thus the next available integer is 4 = a3 , and we delete a3 + 3 = 7, etc.
Thereby we leave the integers 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, . . . . Find a formula for an .
Solution: What we are asking for is a sequence {Sn } which is complementary to the sequence {S√
n + n}. By Beatty’s Theorem,
Tnτ U and Tnτ U + n = Tn(τ + 1)U are complementary if 1/τ + 1/(τ + 1) = 1. But then τ = (1 + 5)/2, the Golden ratio. The
n-th term is thus an = Tnτ U.
Practice
431 Problem (Skolem) Let τ =
√
1+ 5
be the Golden
2
Ratio.
Prove that the three sequences (n ≥ 1)
{Tτ Tτ nUU}, {Tτ Tτ 2 nUU}, {Tτ 2 nU} are complementary.
66
Chapter 6
6.4 Arithmetic Functions
An arithmetic function f is a function whose domain is the set of positive integers and whose range is a subset of the complex
numbers. The following functions are of considerable importance in Number Theory:
d(n)
σ (n)
φ (n)
ω (n)
Ω(n)
the number of positive divisors of the number n.
the sum of the positive divisors of n.
the number of positive integers not exceeding
n and relative prime to n.
the number of distinct prime divisors of n.
the number of primes dividing n, counting multiplicity.
In symbols the above functions are:
d(n) =
X
1, σ (n) =
X
d|n
d, ω (n) =
X
d|n
and
1, Ω(n) =
p|n
X
α,
pα ||n
X
φ (n) =
1.
1≤k≤n
(k,n)=1
(The symbol || in pα ||n is read exactly divides and it signifies that pα |n but pα +1 6 |n.)
For example, since 1, 2, 4, 5, 10 and 20 are the divisors of 20, we have d(20) = 6, σ (20) = 42, ω (20) = 2, Ω(20) = 3. Since
the numbers 1, 3, 7, 9, 11, 13, 17, 19 are the positive integers not exceeding 20 and relatively prime to 20, we see that φ (20) = 8.
If f is an arithmetic function which is not identically 0 such that f (mn) = f (m) f (n) for every pair of relatively prime natural
numbers m, n, we say that f is then a multiplicative function. If f (mn) = f (m) f (n) for every pair of natural numbers m, n we
say then that f is totally multiplicative.
Let f be multiplicative and let n have the prime factorisation n = pa11 pa22 · · · par r . Then
f (n) = f (pa11 ) f (pa22 ) · · · f (par r ).
A multiplicative function is thus determined by its values at prime powers. If f is multiplicative, then there is a positive integer
a such that f (a) 6= 0. Hence f (a) = f (1 · a) = f (1) f (a) which entails that f (1) = 1.
We will now show that the functions d and σ are multiplicative. For this we need first the following result.
432 Theorem Let f be a multiplicative function and let F(n) =
X
f (d). Then F is also multiplicative.
d|n
Proof: Suppose that a, b are natural numbers with (a, b) = 1. By the Fundamental Theorem of Arithmetic, every
divisor d of ab has the form d = d1 d2 where d1 |a, d2 |b, (d1 , d2 ) = 1. Thus there is a one-to-one correspondence
between positive divisors d of ab and pairs d1 , d2 of positive divisors of a and b. Hence, if n = ab, (a, b) = 1 then
F(n) =
X
f (d) =
X X
f (d1 d2 ).
d1 |a d2 |b
d|n
Since f is multiplicative the dextral side of the above equals
X X
d1 |a d2 |b
This completes the proof. ❑
f (d1 ) f (d2 ) =
X
d1 |a
f (d1 )
X
d2 |b
f (d2 ) = F(a)F(b).
Arithmetic Functions
67
Since the function f (n)
= 1 for all natural numbers n is clearly multiplicative (indeed, totally multiplicative), the theorem
X
1 is a multiplicative function. If p is a prime, the divisors of pa are 1, p, p2 , p3 , . . . , pa and so
above shows that d(n) =
d|n
d(pa ) = a + 1. This entails that if n has the prime factorisation n = pa11 pa22 · · · par r , then
d(n) = (1 + a1 )(1 + a2 ) · · · (1 + ar ).
For example, d(2904) = d(23 · 3 · 112 ) = d(23 )d(3)d(112 ) = (1 + 3)(1 + 1)(1 + 2) = 24.
We give now some examples pertaining to the divisor function.
433 Example (AHSME 1993) For how many values of n will an n-sided polygon have interior angles with integral degree
measures?
(n − 2)180
Solution: The measure of an interior angle of a regular n-sided polygon is
. It follows that n must divide 180. Since
n
there are 18 divisors of 180, the answer is 16, because n ≥ 3 and so we must exclude the divisors 1 and 2.
√
434 Example Prove that d(n) ≤ 2 n.
n
n
Solution: Each positive divisor a of n can paired with its complementary divisor . As n = a · , one of these divisors must be
a
a
√
√
≤ n. This gives at most 2 n divisors.
435 Example Find all positive integers n such that d(n) = 6.
Solution: Since 6 can be factored as 2 · 3 and 6 · 1, the desired n must have only two distinct prime factors, p and q, say. Thus
n = pα qβ and either 1 + α = 2, 1 + β = 3 or 1 + α = 6, 1 + β = 1. Hence, n must be of one of the forms pq2 or p5 , where p, q
are distinct primes.
436 Example Prove that
n
X
d(k) =
j=1
k=1
Solution: We have
n
X
d(k) =
X
j≤k≤n
k≡0
1.
k=1 j|k
X
j≤n
n
T U
j
n X
X
k=1
Interchanging the order of summation
n
X
1=
X
j≤n
n
T U,
j
mod j
which is what we wanted to prove.
437 Example (Putnam 1967) A certain locker room contains n lockers numbered 1, 2, . . . , n and are originally locked. An
attendant performs a sequence of operations T1 , T2 , . . . , Tn whereby with the operation Tk , 1 ≤ k ≤ n, the condition of being
locked or unlocked is changed for all those lockers and only those lockers whose numbers are multiples of k. After all the n
operations have been performed it is observed that all lockers whose numbers are perfect squares (and only those lockers) are
now open or unlocked. Prove this mathematically.
Solution: Observe that locker m, 1 ≤ m ≤ n, will be unlocked after n operations if and only if m has an odd number of divisors.
Now, d(m) is odd if and only if m is a perfect square. The assertion is proved.
68
Chapter 6
Since the function f (n) = n is multiplicative (indeed, totally multiplicative), the above theorem entails that σ is multiplicative. If p is a prime, then clearly σ (pa ) = 1+ p+ p2 +· · ·+ pa . This entails that if n has the prime factorisation n = pa11 pa22 · · · par r ,
then
σ (n) = (1 + p1 + p21 + · · · + pa11 )(1 + p2 + p22 + · · · + paw2 ) · · · (1 + pr + p2r + · · · + par r ).
This last product also equals
p1a1 +1 − 1 p2a2 +1 − 1
par +1 − 1
·
··· r
.
p1 − 1
p2 − 1
pr − 1
We present now some examples related to the function σ .
438 Example (Putnam 1969) Let n be a positive integer such that 24|n + 1. Prove that the sum of all divisors of n is also
divisible by 24.
n
Solution: Since 24|n + 1, n ≡ 1 or 2 mod 3 and d ≡ 1, 3, 5 or 7 mod 8. As d( ) ≡ −1 mod 3 or mod 8, the only possibilities
d
are
d ≡ 1, n/d ≡ 2 mod 3 or vice versa,
d ≡ 1, n/d ≡ 7 mod 8 or vice versa,
d ≡ 3, n/d ≡ 5 mod 8 or vice versa.
In all cases d + n/d ≡
0 mod 3 and mod 8, whence 24 divides d + n/d. As d 6≡ n/d, no divisor is used twice in the pairing.
X
d.
This implies that 24|
d|n
We say that a natural number is perfect if it is the sum of its proper divisors. For example, 6 is perfect because 6 =
1 + 2 + 3. It is easy to see that a natural number is perfect if and only if 2n =
X
d=
d|6,d6=6
X
d. The following theorem is classical.
d|n
439 Theorem An even number is perfect if and only if it is of the form 2 p−1 (2 p − 1) where both p and 2 p − 1 are primes.
Proof: Suppose that p, 2 p − 1 are primes. Then σ (2 p − 1) = 1 + 2 p − 1. Since (2 p−1 , 2 p − 1) = 1, σ (2 p−1 (2 p −
1)) = σ (2 p−1 )σ (2 p − 1) = (1 + 2 + 22 + · · · + 2 p−1 )(1 + 2 p − 1) = (2 p − 1)2(2 p−1 ), and 2 p−1 (2 p − 1) is perfect.
Conversely, let n be an even perfect number. Write n = 2s m, m odd. Then σ (n) = σ (2s )σ (m) = (2s+1 − 1)σ (m).
Also, since n perfect is, σ (n) = 2n = 2s+1 m. Hence (2s+1 − 1)σ (m) = 2s+1 m. One deduces that 2s+1 |σ (m), and
so σ (m) = 2s+1 b for some natural number b. But then (2s+1 − 1)b = m, and so b|m, b 6= m.
We propose to show that b = 1. Observe that b + m = (2s+1 − 1)b + b = 2s+1 b = σ (m). If b 6= 1, then there are at
least three divisors of m, namely 1, b and m, which yields σ (m) ≥ 1 + b + m, a contradiction. Thus b = 1, and so
m = (2s+1 − 1)b = 2s+1 − 1 is a prime. This means that 2s+1 − 1 is a Mersenne prime and hence s + 1 must be a
prime.❑
440 Example Prove that for every natural number n there exist natural numbers x and y such that x − y ≥ n and σ (x2 ) = σ (y2 ).
Solution: Let s ≥ n, (s, 10) = 1. We take x = 5s, y = 4s. Then σ (x2 ) = σ (y2 ) = 31σ (s2 ).
Practice
Euler’s Function. Reduced Residues
69
441 Problem Find the numerical values of d(1024), σ (1024), ω (1024),
452
√ Problem Prove that if n is composite, then σ (n) > n +
n.
Ω(1024) and φ (1024).
442 Problem Describe all natural numbers n such that d(n) =
453 Problem Prove that σ (n) = n + k, k > 1 a fixed natural
10.
number has only finitely many solutions.
443 Problem Prove that
454 Problem Characterise all n for which σ (n) is odd.
d(2n − 1) ≥ d(n).
√
444 Problem Prove that d(n) ≤ 3n with equality if and only
if n = 12.
455 Problem Prove that p is a prime if and only if σ (p) =
1 + p.
456 Problem Prove that
1
1
σ (n!)
≥ 1+ +···+ .
n!
2
n
445 Problem Prove that the following Lambert expansion
holds:
∞
X
d(n)t n =
n=1
∞
X
n=1
tn
.
1 − tn
457 Problem Prove that an odd perfect number must have at
least two distinct prime factors.
446 Problem Let d1 (n) = d(n), dk (n) = d(dk−1 (n)), k =
2, 3, . . .. Describe dk (n) for sufficiently large k.
447 Problem Let m ∈ N be given. Prove that the set
A = {n ∈ N : m|d(n)}
contains an infinite arithmetic progression.
458 Problem Prove that in an odd perfect number, only one
of its prime factors occurs to an odd power; all the others occur
to an even power.
459 Problem Show that an odd perfect number must contain
one prime factor p such that, if the highest power of p occurring in n is pa , both p and a are congruent to 1 modulo 4; all
other prime factors must occur to an even power.
448 Problem Let n be a perfect number. Show that
X
d|n
460 Problem Prove that every odd perfect number having
1
= 2.
d
three distinct prime factors must have two of its prime factors
3 and 5.
449 Problem Prove that
Y
461 Problem Prove that there do not exist odd perfect numd(n)/2
d=n
bers having exactly three distinct prime factors.
.
d|n
462 Problem Prove that
450 Problem Prove that the power of a prime cannot be a per-
n
X
fect number.
k=1
σ (k) =
n
X
j=1
n
jT U.
j
31 19
451 Problem (AIME, 1995) Let n = 2 3 . How many positive integer divisors of n2 are less than n but do not divide
n?
463 Problem Find the number of sets of positive integers
{a, b, c} such that a × b × c = 462.
6.5 Euler’s Function. Reduced Residues
Recall that Euler’s φ (n) function counts the number of positive integers a ≤ n that are relatively prime to n. We will prove now
that φ is multiplicative. This requires more work than that done for d and σ .
First we need the following definitions.
70
Chapter 6
464 Definition Let n > 1. The φ (n) integers 1 = a1 < a2 < · · · < aφ (n) = n − 1 less than n and relatively prime to n are called
the canonical reduced residues modulo n.
465 Definition A reduced residue system modulo n, n > 1 is a set of φ (n) incongruent integers modulo n that are relatively
prime to n.
For example, the canonical reduced residues mod 12 are 1, 5, 7, 11 and the set {−11, 5, 19, 23} forms a reduced residue
system modulo 12.
We are now ready to prove the main result of this section.
466 Theorem The function φ is multiplicative.
Proof: Let n be a natural number with n = ab, (a, b) = 1. We arrange the ab integers 1, 2, . . . , ab as follows.
1
2
3
a+1
a+2
a+3
2a + 1
2a + 2
2a + 3
...
...
...
(b − 1)a + 1 (b − 1)a + 2 (b − 1)a + 3
...
k
...
a+k
...
2a + k
...
...
. . . (b − 1)a + k
...
a
. . . 2a
. . . 3a
... ...
. . . ba
Now, an integer r is relatively prime to m if and only if it is relatively prime to a and b. We shall determine first the
number of integers in the above array that are relatively prime to a and find out how may of them are also relatively
prime to b.
There are φ (a) integers relatively prime to a in the first row. Now consider the k-th column, 1 ≤ k ≤ a. Each integer
on this column is of the form ma + k, 0 ≤ m ≤ b − 1. As k ≡ ma + k mod a, k will have a common factor with a if
and only if ma + k does. This means that there are exactly φ (a) columns of integers that are relatively prime to a.
We must determine how many of these integers are relatively prime to b.
We claim that no two integers k, a + k, . . . , (b − 1)a + k on the k-th column are congruent modulo b. For if ia + k ≡
ja + k mod b then a(i − j) ≡ 0 mod b. Since (a, b) = 1, we deduce that i − j ≡ 0 mod b thanks to Corollary 346.
Now i, j ∈ [0, b − 1] which implies that |i − j| < b. This forces i = j. This means that the b integers in any of these
φ (n) columns are, in some order, congruent to the integers 0, 1, . . . , b − 1. But exactly φ (b) of these are relatively
prime to b. This means that exactly φ (a)φ (b) integers on the array are relatively prime to ab, which is what we
wanted to show.❑
If p is a prime and m a natural number, the integers
p, 2p, 3p, . . . , pm−1 p
are the only positive integers ≤ pm sharing any prime factors with pm . Thus φ (pm ) = pm − pm−1 . Since φ is multiplicative, if
a
n = pa11 · · · pk k is the factorisation of n into distinct primes, then
a −1
φ (n) = (pa11 − pa11 −1 ) · · · (pk k − pk k
a
).
For example, φ (48) = φ (24 · 3) = φ (24 )φ (3) = (24 − 23 )(3 − 1) = 16, and φ (550) = φ (2 · 52 · 11) = φ (2) · φ (52 ) · φ (11) =
(2 − 1)(52 − 5)(11 − 1) = 1 · 20 · 10 = 200.
467 Example Let n be a natural number. How many of the fractions 1/n, 2/n, . . . , (n − 1)/n, n/n are irreducible?
n
X
Solution: This number is clearly
k=1
φ (k).
Euler’s Function. Reduced Residues
71
468 Example Prove that for n > 1,
X
a=
1≤a≤n
nφ (n)
.
2
(a,n)=1
Solution: Clearly if 1 ≤ a ≤ n and (a, n) = 1, 1 ≤ n − a ≤ n and (n − a, n) = 1. Thus
X
S=
a=
X
1≤a≤n
1≤a≤n
(a,n)=1
(a,n)=1
whence
X
2S =
n − a,
n = nφ (n).
1≤a≤n
(a,n)=1
The assertion follows.
X
469 Theorem Let n be a positive integer. Then
φ (d) = n.
d|n
Proof: For each divisor d of n, let Td (n) be the set of positive integers ≤ n whose gcd with n is d. As d varies
over the divisors of n, the Td partition the set {1, 2, . . . , n} and so
X
Td (n) = n.
d|n
We claim that Td (n) has φ (n/d) elements. Note that the elements of Td (n) are found amongst the integers
k n
n
n
d, 2d, . . . d. If k ∈ Td (n), then k = ad, 1 ≤ a ≤ n/d and (k, n) = d. But then ( , ) = 1. This implies that (a, ) = 1.
d
d d
d
n
Therefore counting the elements of Td (n) is the same as counting the integers a with 1 ≤ a ≤ n/d, (a, ) = 1. But
d
there are exactly φ (n/d) such a. We gather that
n=
X
φ (n/d).
d|n
But as d runs through the divisors of n, n/d runs through the divisors of n in reverse order, whence n =
X
d|n
φ (d).❑
d|n
470 Example If p − 1 and p + 1 are twin primes, and p > 4, prove that 3φ (p) ≤ p.
Solution: Observe that p > 4 must be a multiple of 6, so
p = 2a 3b m, ab ≥ 1, (m, 6) = 1.
We then have φ (p) ≤ 2a 3b−1 φ (m) ≤ 2a 3b−1 m = p/3.
471 Example Let n ∈ N. Prove that the equation
is soluble.
X
φ (x) = n!
φ (n/d) =
72
Chapter 6
Solution: We want to solve the equation φ (x) = n with the constraint that x has precisely the same prime factors as n. This
restriction implies that φ (x)/x = φ (n)/n. It follows that x = n2 /φ (n).
Let n =
Y
pα . Then x =
pα ||n
Y
pα ||n
Y
pα
. The integer x will have the same prime factors as n provided that
(p − 1)|n. It is
p−1
p|n
clear then that a necessary
and sufficient condition for φ (x) = n to be soluble under the restriction that x has precisely the same
Y
(p − 1)|n. If n = k!, this last condition is clearly satisfied. An explicit solution to the problem is thus
prime factors as n is
p|n
x = (k!)2 /φ (k!).
472 Example Let φk (n) = φ (φk−1 (n)), k = 1, 2, . . . , where φ0 (n) = φ (n). Show that ∀ k ∈ N, φk (n) > 1 for all sufficiently large
n.
Solution: Let pa11 pa22 · · · par r be the prime factorisation of n. Clearly
a /2 a /2
a /2
p11 p22 · · · pr r
Hence
φ (n) =
> 2r−1 ≥
1 p1
pr
···
.
2 p1 − 1
pr − 1
a a
p1 − 1 p2 − 1
pr − 1 a1 a2
1 p11 p22 · · · par r
.
···
p1 p2 · · · par r ≥
p1
p2
pr
2 pa1 /2 pa2 /2 · · · prar /2
1
2
√
1È
1
This last quantity equals n/2. Therefore φ1 (n) >
φ (n) >
2
2
k+2
1 2−k−1
n
. We conclude that n ≥ 22 implies that φk (n) > 1.
4
r
1√
1
n = n1/4 . In general we can show that φk (n) >
4
4
473 Example Find infinitely many integers n such that 10|φ (n).
Solution: Take n = 11k , k = 1, 2, . . .. Then φ (11k ) = 11k − 11k−1 = 10 · 11k−1 .
Practice
479 Problem If φ (n)|n, then n must be of the form 2a 3b for
474 Problem Prove that
φ (n) = n
Y

p|n
nonnegative integers a, b.
‹
1
1−
.
p
475 Problem Prove that if n is composite then φ (n) ≤ n−
When is equality achieved?
480 Problem Prove that if φ (n)|n − 1, then n must be square-
√
n.
free.
481 Problem (Mandelbrot 1994) Four hundred people are
standing in a circle. You tag one person, then skip k people,
then tag another, skip k, and so on, continuing until you tag
476 Problem (AIME 1992) Find the sum of all positive rasomeone for the second time. For how many positive values
tional numbers that are less than 10 and have denominator 30
of k less than 400 will every person in the circle get tagged at
when written in lowest terms.
least once?
Answer: 400
477 Problem Prove that φ (n) ≥ n2−ω (n) .
478 Problem Prove that φ (n) >
√
n for n > 6.
482 Problem Prove that if φ (n)|n−1 and n is composite, then
n has at least three distinct prime factors.
483 Problem Prove that if φ (n)|n−1 and n is composite, then
n has at least four prime factors.
Multiplication in Zn
73
484 Problem For n > 1 let 1 = a1 < a2 < · · · < aφ (n) = n − 1
(Hint: Use the Chinese Remainder Theorem).
be the positive integers less than n that are relatively prime to
n. Define the Jacobsthal function
g(n) :=
max
1≤k≤φ (n)−1
485 Problem Prove that a necessary and sufficient condition
for n to be a prime is that
ak+1 − ak
to be the maximum gap between the ak . Prove that ω (n) ≤
g(n).
σ (n) + φ (n) = nd(n).
6.6 Multiplication in Zn
In section 3.5 we saw that Zn endowed with the operation of addition +n becomes a group. We are now going to investigate the
multiplicative structure of Zn .
How to define multiplication in Zn ? If we want to multiply a ·n b we simply multiply a · b and reduce the result mod n. As
an example, let us consider Table 6.1. To obtain 4 ·6 2 we first multiplied 4 · 2 = 8 and then reduced mod 6 obtaining 8 ≡ 2
mod 6. The answer is thus 4 ·6 2 = 2.
Another look at the table shows the interesting product 3 ·6 2 = 0. Why is it interesting? We have multiplied to non-zero
entities and obtained a zero entity!
Does Z6 form a group under ·6 ? What is the multiplicative identity? In analogy with the rational numbers, we would like
1 to be the multiplicative identity. We would then define the multiplicative inverse of a to be that b that has the property that
a ·6 b = b ·6 a = 1. But then, we encounter some problems. For example, we see that 0, 2, 3, and 4 do not have a multiplicative
inverse. We need to be able to identify the invertible elements of Zn . For that we need the following.
·6
0
1
2
3
4
5
0
0
0
0
0
0
0
1
0
1
2
3
4
5
2
0
2
4
0
2
4
3
0
3
0
3
0
3
4
0
4
2
0
4
2
5
0
5
4
3
2
1
Table 6.1: Multiplication Table for Z6
486 Definition Let n > 1 be a natural number. An integer b is said to be the inverse of an integer a modulo n if ab ≡ 1 mod n.
It is easy to see that inverses are unique mod n. For if x, y are inverses to a mod n then ax ≡ 1 mod n and ay ≡ 1 mod n.
Multiplying by y the first of these congruences, (ya)x ≡ y mod n. Hence x ≡ y mod n.
487 Theorem Let n > 1, a be integers. Then a possesses an inverse modulo n if and only if a is relatively prime to n.
Proof: Assume that b is the inverse of a mod n. Then ab ≡ 1 mod n, which entails the existence of an integer s
such that ab − 1 = sn, i.e. ab − sn = 1. This is a linear combination of a and n and hence divisible by (a, n). This
implies that (a, n) = 1.
Conversely if (a, n) = 1, by the Bachet-Bezout Theorem there are integers x, y such that ax + ny = 1. This immediately yields ax ≡ 1 mod n, i.e., a has an inverse mod n.❑
488 Example Find the inverse of 5 mod 7.
Solution: We are looking for a solution to the congruence 5x ≡ 1 mod 7. By inspection we see that this is x ≡ 3 mod 7.
74
Chapter 6
According to the preceding theorem, a will have a multiplicative inverse if and only if (a, n) = 1. We thus see that only the
reduced residues mod n have an inverse. We let Z×
n = {a1 , a2 , . . . , aφ (n) }. It is easy to see that the operation ·n is associative,
since it inherits associativity from the integers. We conclude that Z×
n is a group under the operation ·n .
We now give some assorted examples.
489 Example (IMO 1964) Prove that there is no positive integer n for which 2n + 1 is divisible by 7.
Solution: Observe that 21 ≡ 2, 22 ≡ 4, 23 ≡ 1 mod 7, 24 ≡ 2 mod 7, 25 ≡ 4 mod 7, 26 ≡ 1 mod 7, etc. The pattern 2, 4, 1,
repeats thus cyclically. This says that there is no power of 2 which is ≡ −1 ≡ 6 mod 7.
490 Theorem If a is relatively prime to the positive integer n, there exists a positive integer k ≤ n such that ak ≡ 1 mod n.
Proof: Since (a, n) = 1 we must have (a j , n) = 1 for all j ≥ 1. Consider the sequence a, a2 , a3 , . . . , an+1 mod n.
As there are n + 1 numbers and only n residues mod n, the Pigeonhole Principle two of these powers must have
the same remainder mod n. That is, we can find s,t with 1 ≤ s < t ≤ n + 1 such that as ≡ at mod n. Now,
1 ≤ t − s ≤ n. Hence as ≡ at mod n gives at−s as ≡ at−s at mod n, which is to say at ≡ at−s at mod n. Using
Corollary 346 we gather that at−s ≡ 1 mod n, which proves the result.❑
If (a, n) = 1, the preceding theorem tells us that there is a positive integer k with ak ≡ 1 mod n. By the Well-Ordering
Principle, there must be a smallest positive integer with this property. This prompts the following definition.
491 Definition If m is the least positive integer with the property that am ≡ 1 mod n, we say that a has order m mod n.
For example, 31 ≡ 3, 32 ≡ 2, 33 ≡ 6, 34 ≡ 4, 35 ≡ 5, 36 ≡ 1 mod 7, and so the order of 3 mod 7 is 6. We write this fact as
ord7 3 = 6.
Given n, not all integers a are going to have an order mod n. This is clear if n|a, because then am ≡ 0 mod n for all
positive integers m. The question as to which integers are going to have an order mod n is answered in the following theorem.
492 Theorem Let n > 1 be a positive integer. Then a ∈ Z has an order mod n if and only if (a, n) = 1.
Proof: If (a, n) = 1, then a has an order in view of Theorem 490 and the Well-Ordering Principle. Hence assume
that a has an order mod n. Clearly a 6= 0. The existence of an order entails the existence of a positive integer
m such that am ≡ 1 mod n. Hence, there is an integer s with am + sn = 1 or a · am−1 + sn = 1. This is a linear
combination of a and n and hence divisible by (a, n). This entails that (a, n) = 1. ❑
The following theorem is of utmost importance.
493 Theorem Let (a, n) = 1 and let t be an integer. Then at ≡ 1 mod n if and only if ordn a|t.
Proof: Assume that ordn a|t. Then there is an integer s such that sordn a = t. This gives
at ≡ asordn a ≡ (aordn a )s ≡ 1s ≡ 1
mod n.
Conversely, assume that at ≡ 1 mod n and t = x · ordn a + y, 0 ≤ y < ordn a. Then
ay ≡ at−xordn a ≡ at · (aordn a )−x ≡ 1 · 1−x ≡ 1
mod n.
If y > 0 we would have a positive integer smaller than ordn a with the property ay ≡ 1 mod n. This contradicts
the definition of ordn a as the smallest positive integer with that property. Hence y = 0 and thus t = x · ordn a, i.e.,
ordn a|t.❑
Practice
75
494 Example (IMO 1964) Find all positive integers n for which 2n − 1 is divisible by 7.
Solution: Observe that the order of 2 mod 7 is 3. We want 2n ≡ 1 mod 7. It must then be the case that 3|n. Thus n =
3, 6, 9, 12, . . ..
The following result will be used repeatedly.
495 Theorem Let n > 1, a ∈ Z, (a, n) = 1. If r1 , r2 , . . . , rφ (n) is a reduced set of residues modulo n, then ar1 , ar2 , . . . , arφ (n) is
also a reduced set of residues modulo n.
Proof: We just need to show that the φ (n) numbers ar1 , ar2 , . . . , arφ (n) are mutually incongruent mod n. Suppose
that ari ≡ ar j mod n for some i 6= j. Since (a, n) = 1, we deduce from Corollary 346 that ri ≡ r j mod n. This
contradicts the fact that the r’s are incongruent, so the theorem follows.❑
For example, as 1, 5, 7, 11 is a reduced residue system modulo 12 and (12, 5) = 1, the set 5, 25, 35, 55 is also a reduced
residue system modulo 12. Again, the 1, 5, 7, 11 are the 5, 25, 35, 55 in some order and 1 · 5 · 7 · 11 ≡ 5 · 25 · 35 · 55 mod 12.
The following corollary to Theorem 495 should be immediate.
496 Corollary Let n > 1, a, b ∈ Z, (a, n) = 1. If r1 , r2 , . . . , rφ (n) is a reduced set of residues modulo n, then ar1 + b, ar2 +
b, . . . , arφ (n) + b is also a reduced set of residues modulo n.
Practice
497 Problem Find the order of 5 modulo 12.
6.7 Möbius Function
498 Definition The Möbius function is defined for positive integer n as follows:
8
µ (n) =
<
:
if n = 1,
if ω (n) = Ω(n),
if ω (n) < Ω(n).
1
(−1)ω (n)
0
Thus µ is 1 for n = 1 and square free integers with an even number of prime factors, −1 for square free integers with an
odd number of prime factors, and 0 for non-square free integers. Thus for example µ (6) = 1, µ (30) = −1 and µ (18) = 0.
499 Theorem The Möbius Function µ is multiplicative.
Proof: Assume (m, n) = 1. If both M and n are square-free then
µ (m)µ (n) = (−1)ω (m) (−1)ω (n) = (−1)ω (m)+ω (n) = µ (mn).
If one of m, n is not square-free then
µ (m)µ (n) = 0 = µ (mn).
This proves the theorem. ❑
500 Theorem
X
d|n
§
µ (d) =
1
0
if n = 1,
if n > 1.
76
Chapter 6
ω (n)
Proof: There are
square-free divisors d of n with exactly k prime factors. For all such d, µ (d) = (−1)k .
k
The sum in question is thus
X
ω (n) µ (d) =
X
k=0
d|n
ω (n)
(−1)k .
k
By the Binomial Theorem this last sum is (1 − 1)ω (n) = 0.❑
501 Theorem (Möbius Inversion Formula) Let f be an arithmetical function and F(n) =
X
f (d). Then
d|n
f (n) =
X
µ (d)F(n/d) =
X
d|n
Proof: We have
X
µ (n/d)F(d).
d|n
µ (d)F(n/d) =
d|n
XXX
f (s)
n
s|
d
X
µ (d) f (s)
d|n d|n
=
ds|n
X
X
µ (d).
n
d|
s
n
In view of theorem 500, the inner sum is different from 0 only when = 1. Hence only the term s = n in the outer
s
sum survives, which means that the above sums simplify to f (n).❑
=
f (s)
s|n
We now show the converse to Theorem 501.
502 Theorem Let f , F be arithmetic functions with f (n) =
X
µ (d)F(n/d) for all natural numbers n. Then F(n) =
d|n
Proof: We have
X
f (d) =
d|n
X
f (d).
d|n
XX
µ (s)F(d/s)
d|n s|d
=
XX
µ (d/s)F(s)
d|n s|d
=
XX
s|n
µ (r)F(s).
n
r|
s
Using Theorem 500, the inner sum will be 0 unless s = n, in which case the entire sum reduces to F(n).❑
Practice
504 Problem If f is an arithmetical function and F(n) =
503 Problem Prove that
φ (n) = n
X
d|n
µ (d)
.
d
Practice
n
X
77
X
506 Problem Prove that
f ([n/k]), then
d|n
k=1
f (n) =
n
X
µ ( j)F([n/ j]).
505 Problem If F is an arithmetical function such that f (n) =
k=1
µ (k)F([n/k]), prove that F(n) =
µ (d)d(d) = (−1)ω (n) .
d|n
j=1
n
X
X
507 Problem Prove that
|µ (d)| = 2ω (n) .
n
X
j=1
f ( j).
508 Problem Given any positive integer k, prove that there
exist infinitely many integers n with
µ (n + 1) = µ (n + 2) = · · · = µ (n + k).
Chapter
7
More on Congruences
7.1 Theorems of Fermat and Wilson
509 Theorem (Fermat’s Little Theorem) Let p be a prime and let p 6 |a. Then
a p−1 ≡ 1
mod p.
Proof: Since (a, p) = 1, the set a · 1, a · 2, . . . , a · (p − 1) is also a reduced set of residues
Theorem 495. Hence
(a · 1)(a · 2) · · · (a · (p − 1)) ≡ 1 · 2 · · · (p − 1) mod p,
mod p in view of
or
a p−1 (p − 1)! ≡ (p − 1)!
mod p.
As ((p − 1)!, p) = 1 we may cancel out the (p − 1)!’s in view of Corollary 346. This proves the theorem.❑
As an obvious corollary, we obtain the following.
510 Corollary For every prime p and for every integer a,
ap ≡ a
mod p.
Proof: Either p|a or p 6 |a. If p|a, a ≡ 0 ≡ a p mod p and there is nothing to prove. If p 6 |a, Fermat’s Little
Theorem yields p|a p−1 − 1. Hence p|a(a p−1 − 1) = a p − a, which again gives the result.❑
The following corollary will also be useful.
511 Corollary Let p be a prime and a an integer. Assume that p 6 |a. Then ord p a|p − 1.
Proof: This follows immediately from Theorem 493 and Fermat’s Little Theorem.❑
512 Example Find the order of 8 mod 11.
Solution: By Corollary 511 ord11 8 is either 1, 2, 5 or 10. Now 82 ≡ −2 mod 11, 84 ≡ 4 mod 11 and 85 ≡ −1 mod 11. The
order is thus ord11 8 = 10.
513 Example Let a1 = 4, an = 4an−1 , n > 1. Find the remainder when a100 is divided by 7.
78
Theorems of Fermat and Wilson
79
Solution: By Fermat’s Little Theorem, 46 ≡ 1 mod 7. Now, 4n ≡ 4 mod 6 for all positive integers n, i.e., 4n = 4 + 6t for some
integer t. Thus
a100 ≡ 4a99 ≡ 44+6t ≡ 44 · (46 )t ≡ 4 mod 7.
514 Example Prove that for m, n ∈ Z, mn(m60 − n60 ) is always divisible by 56786730.
Solution: Let a = 56786730 = 2 · 3 · 5 · 7 · 11 · 13 · 31 · 61. Let Q(x, y) = xy(x60 − y60 ). Observe that (x − y)|Q(x, y), (x2 −
y2 )|Q(x, y), (x3 − y3 )|Q(x, y), (x4 − y4 )|Q(x, y), (x6 − y6 )|Q(x, y), (x10 − y10 )|Q(x, y), (x12 − y12 )|Q(x, y), and (x30 − y30 )|Q(x, y).
If p is any one of the primes dividing a, the Corollary to Fermat’s Little Theorem yields m p − m ≡ 0 mod p and n p −
n ≡ 0 mod p. Thus n(m p − m) − m(n p − n) ≡ 0 mod p, i.e., mn(m p−1 − n p−1 ) ≡ 0 mod p. Hence, we have 2|mn(m −
n)|Q(m, n), 3|mn(m2 −n2 )|Q(m, n), 5|mn(m4 −n4 )|Q(m, n), 7|mn(m6 −n6 )|Q(m, n), 11|mn(m10 −n10 )|Q(m, n), 13|mn(m12 −n12 )|Q(m, n), 31|m
n30 )|Q(m, n) and 61|mn(m60 − n60 )|Q(m, n). Since these are all distinct primes, we gather that a|mnQ(m, n), which is what we
wanted.
515 Example (Putnam 1972) Show that given an odd prime p, there are always infinitely many integers n for which p|n2n + 1.
Answer: For any odd prime p, take n = (p − 1)2k+1 , k = 0, 1, 2, . . .. Then
2k
n2n + 1 ≡ (p − 1)2k+1 (2 p−1 )(p−1) + 1 ≡ (−1)2k+1 12k + 1 ≡ 0
mod p.
516 Example Prove that there are no integers n > 1 with n|2n − 1.
Solution: If n|2n − 1 for some n > 1, then n must be odd and have a smallest odd prime p as a divisor. By Fermat’s Little
Theorem, 2 p−1 ≡ 1 mod p. By Corollary 511 , ord p 2 has a prime factor in common with p − 1. Now, p|n|2n − 1 and so 2n ≡ 1
mod p. Again, by Corollary 511, ord p 2 must have a common prime factor with n (clearly ord p 2 > 1). This means that n has a
smaller prime factor than p, a contradiction.
517 Example Let p be a prime. Prove that
1.
2.
p−1
≡ (−1)n
n
mod p, 1 ≤ n ≤ p − 1.
p+1
≡0
n
mod p, 2 ≤ n ≤ p − 1.
3. If p 6= 5 is an odd prime, prove that either f p−1 or f p+1 is divisible by p.
Solution: (1) (p − 1)(p − 2) · · · (p − n) ≡ (−1)(−2) · · · (−n) ≡ (−1)n n! mod p. The assertion follows from this.
(2) (p + 1)(p)(p − 1) · · · (p − n + 2) ≡ (1)(0)(−1) · · · (−n + 2) ≡ 0 mod p. The assertion follows from this.
(3) Using the Binomial Theorem and Binet’s Formula
fn =
1
2n−1
 ‹
n
n
n
+5
+ 52
+··· .
1
3
5
From this and (1),
2 p−2 f p−1 ≡ p − 1 − (5 + 52 + · · · + 5(p−3)/2 ) ≡ −
5(p−1)/2 − 1
4
Using (2),
2 p f p+1 ≡ p + 1 + 5(p−1)/2 ≡ 5(p−1)/2 + 1
mod p.
mod p.
80
Chapter 7
Thus
But by Fermat’s Little Theorem, 5
p−1
2 p f p−1 f p+1 ≡ 5 p−1 − 1
mod p.
≡ 1 mod p for p 6= 5. The assertion follows.
518 Lemma If a2 ≡ 1 mod p, then either a ≡ 1 mod p or a ≡ −1 mod p.
Proof: We have p|a2 − 1 = (a − 1)(a + 1). Since p is a prime, it must divide at least one of the factors. This
proves the lemma.❑
519 Theorem (Wilson’s Theorem) If p is a prime, then (p − 1)! ≡ −1 mod p.
Proof: If p = 2 or p = 3, the result follows by direct verification. So assume that p > 3. Consider a, 2 ≤ a ≤ p − 2.
To each such a we associate its unique inverse a mod p, i.e. aa ≡ 1 mod p. Observe that a 6= a since then we
would have a2 ≡ 1 mod p which violates the preceding lemma as a 6= 1, a 6= p − 1. Thus in multiplying all a in the
range 2 ≤ a ≤ p − 2, we pair them of with their inverses, and the net contribution of this product is therefore 1. In
symbols,
2 · 3 · · · (p − 2) ≡ 1 mod p.
In other words,
„
(p − 1)! ≡ 1 ·
Ž
Y
· (p − 1) ≡ 1 · 1 · (p − 1) ≡ −1
j
2≤a≤p−2
mod p.
This gives the result. ❑
520 Example If p ≡ 1 mod 4, prove that

‹
p−1
! ≡ −1
2
mod p.
Solution: In the product (p − 1)! we pair off j, 1 ≤ j ≤ (p − 1)/2 with p − j. Observe that j(p − j) ≡ − j2 mod p. Hence
−1 ≡ (p − 1)! ≡
(p−1)/2
As (−1)
Y
1≤ j≤(p−1)/2

− j2 ≡ (−1)(p−1)/2
‹
p−1
!
2
mod p.
= 1, we obtain the result.
521 Example (IMO 1970) Find the set of all positive integers n with the property that the set
{n, n + 1, n + 2, n + 3, n + 4, n + 5}
can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other
set.
Solution: We will show that no such partition exists. Suppose that we can have such a partition, with one of the subsets having
product of its members equal to A and the other having product of its members equal to B. We might have two possibilities.
The first possibility is that exactly one of the numbers in the set {n, n + 1, n + 2, n + 3, n + 4, n + 5} is divisible by 7, in which
case exactly one of A or B is divisible by 7, and so A · B is not divisible by 72 , and so A · B is not a square. The second possibility
is that all of the members of the set are relatively prime to 7. In this last case we have
n(n + 1) · · · (n + 6) ≡ 1 · 2 · · · 6 ≡ A · B ≡ −1
2
mod 7.
But if A = B then we are saying that there is an integer A such that A ≡ −1 mod 7, which is an impossibility, as −1 is not a
square mod 7. This finishes the proof.
Practice
Euler’s Theorem
81
522 Problem Find all the natural numbers n for which
528 Problem Let p and q be distinct primes. Prove that
3|(n2n + 1).
q p−1 + pq−1 ≡ 1
mod pq.
523 Problem Prove that there are infinitely many integers n
529 Problem If p is an odd prime prove that n p ≡ n mod 2p
with n|2n + 2.
for all integers n.
p
524 Problem Find all primes p such that p|2 + 1.
530 Problem If p is an odd prime and p|m p + n p prove that
p2 |m p + n p .
Answer: p = 3 only.
525 Problem If p and q are distinct primes prove that
pq|(a pq − a p − aq − a)
531 Problem Prove that n > 1 is a prime if and only if
(n − 1)! ≡ −1 mod n.
for all integers a.
532 Problem Prove that if p is an odd prime
526 Problem If p is a prime prove that p|a p + (p − 1)!a for
12 ·32 · · · (p−2)2 ≡ 22 ·42 · · · (p−1)2 ≡ (−1)(p−1)/2
all integers a.
533 Problem Prove that 19|(22
6
6
527 Problem If (mn, 42) = 1 prove that 168|m − n .
6k+2
mod p
+ 3) for all nonnegative
integers k.
7.2 Euler’s Theorem
In this section we obtain a generalisation of Fermat’s Little Theorem, due to Euler. The proof is analogous to that of Fermat’s
Little Theorem.
534 Theorem (Euler’s Theorem) Let (a, n) = 1. Then aφ (n) ≡ 1 mod n.
Proof: Let a1 , a2 , . . . , aφ (n) be the canonical reduced residues mod n. As (a, n) = 1, aa1 , aa2 , . . . , aaφ (n) also
forms a set of incongruent reduced residues. Thus
or
aa1 · aa2 · · · aaφ (n) ≡ a1 a2 · · · aφ (n)
mod n,
aφ (n) a1 a2 · · · aφ (n) ≡ a1 a2 · · · aφ (n)
mod n.
As (a1 a2 · · · aφ (n) , n) = 1, we may cancel the product a1 a2 · · · aφ (n) from both sides of the congruence to obtain
Euler’s Theorem.❑
Using Theorem 534 we obtain the following corollary.
535 Corollary Let (a, n) = 1. Then ordn a|φ (n).
536 Example Find the last two digits of 31000 .
Solution: As φ (100) = 40, by Euler’s Theorem, 340 ≡ 1 mod 100. Thus
31000 = (340 )25 ≡ 125 = 1
and so the last two digits are 01.
537 Example Find the last two digits of 77
1000
.
mod 100,
82
Chapter 7
Solution: First observe that φ (100) = φ (22 )φ (52 ) = (22 − 2)(52 − 5) = 40. Hence, by Euler’s Theorem, 740 ≡ 1 mod 100.
Now, φ (40) = φ (23 )φ (5) = 4 · 4 = 16, hence 716 ≡ 1 mod 40. Finally, 1000 = 16 · 62 + 8. This means that 71000 ≡ (716 )62 78 ≡
162 78 ≡ (74 )2 ≡ 12 ≡ 1 mod 40. This means that 71000 = 1 + 40t for some integer t. Upon assembling all this
77
1000
≡ 71+40t ≡ 7 · (740 )t ≡ 7
mod 100.
This means that the last two digits are 07.
538 Example (IMO 1978) m, n are natural numbers with 1 ≤ m < n. In their decimal representations, the last three digits of
1978m are equal, respectively, to the last three digits of 1978n . Find m, n such that m + n has its least value.
Solution: As m + n = n − m + 2m, we minimise n − m. We are given that
1978n − 1978m = 1978m (1978n−m − 1)
is divisible by 1000 = 23 53 . Since the second factor is odd, 23 must divide the first and so m ≥ 3. Now, ord125 1978 is the
smallest positive integer s with
1978s ≡ 1 mod 125.
By Euler’s Theorem
s
1978100 ≡ 1
mod 125
and so by Corollary 7.3 s|100. Since 125|(1978 − 1) we have 5|(1978s − 1), i.e., 1978s ≡ 3s ≡ 1 mod 5. Since s|100, this last
congruence implies that s = 4, 20, or 100. We now rule out the first two possibilities.
Observe that
19784 ≡ (−22)4 ≡ 24 · 114 ≡ (4 · 121)2 ≡ (−16)2 ≡ 6
mod 125.
This means that s 6= 4. Similarly
197820 ≡ 19784 · (19784 )4 ≡ 6 · 64 ≡ 6 · 46 ≡ 26
mod 125.
This means that s 6= 20 and so s = 100. Since s is the smallest positive integer with 1978s ≡ 1 mod 125, we take n−m = s = 100
and m = 3, i.e., n = 103, m = 3, and finally, m + n = 106.
539 Example (IMO 1984) Find one pair of positive integers a, b such that:
(i) ab(a + b) is not divisible by 7.
(ii) (a + b)7 − a7 − b7 is divisible by 77 . Justify your answer.
Solution: We first factorise (a + b)7 − a7 − b7 as ab(a + b)(a2 + ab + b2 )2 . Using the Binomial Theorem we have
(a + b)7 − a7 − b7
=
=
=
=
=
7(a6 b + ab6 + 3(a5 b2 + a2 b5 ) + 5(a4 b3 + a3 b4 ))
7ab(a5 + b5 + 3ab(a3 + b3 ) + 5(a2 b2 )(a + b))
7ab(a + b)(a4 + b4 − a3 b − ab3 + a2 b2
+3ab(a2 − ab + b2 ) + 5ab)
7ab(a + b)(a4 + b4 + 2(a3 b + ab3 ) + 3a2 b2 )
7ab(a + b)(a2 + ab + b2 )2 .
The given hypotheses can be thus simplified to
(i)′ ab(a + b) is not divisible by 7,
(ii)′ a2 + ab + b2 is divisible by 73 .
As (a + b)2 > a2 + ab + b2 ≥ 73 , we obtain a + b ≥ 19. Using trial and error, we find that a = 1, b = 18 give an answer, as
12 + 1 · 18 + 182 = 343 = 73 .
Practice
83
Let us look for more solutions by means of Euler’s Theorem. As a3 − b3 = (a − b)(a2 + ab + b2 ), (ii)’ is implied by
′′
(ii)
a3 ≡ b3 mod 73
a 6≡ b mod 7.
Now φ (73 ) = (7 − 1)72 = 3 · 98, and so if x is not divisible by 7 we have (x98 )3 ≡ 1 mod 73 , which gives the first part of (ii)’.
We must verify now the conditions of non-divisibility. For example, letting x = 2 we see that 298 ≡ 4 mod 7. Thus letting
a = 298 , b = 1. Letting x = 3 we find that 398 ≡ 324 mod 73 . We leave to the reader to verify that a = 324, b = 1 is another
solution.
Practice
540 Problem Show that for all natural numbers s, there is an 546 Problem Find the last two digits of a1001 if a1 = 7, an =
integer n divisible by s, such that the sum of the digits of n 7an−1 .
equals s.
541 Problem Prove that 504|n9 − n3 .
547 Problem Find the remainder of
2
542 Problem Prove that for odd integer n > 0, n|(2n! − 1).
1010 + 1010 + · · · + 1010
10
upon division by 7.
543 Problem Let p 6 |10 be a prime. Prove that p divides infinitely many numbers of the form
548 Problem Prove that for every natural number n there ex-
ists some power of 2 whose final n digits are all ones and twos.
11 . . . 11.
544 Problem Find all natural numbers n that divide
1n + 2n + · · · + (n − 1)n .
549 Problem (USAMO 1982) Prove that there exists a positive integer k such that k · 2n + 1 is composite for every positive
integer n.
545 Problem Let (m, n) = 1. Prove that
mφ (n) + nφ (n) ≡ 1
mod mn.
550 Problem (Putnam 1985) Describe the sequence a1 =
3, an = 3an−1 mod 100 for large n.
Chapter
8
Scales of Notation
8.1 The Decimal Scale
As we all know, any natural number n can be written in the form
n = a0 10k + a1 10k−1 + · · · + ak−1 10 + ak ,
where 1 ≤ a0 ≤ 9, 0 ≤ a j ≤ 9, j ≥ 1. For example, 65789 = 6 · 104 + 5 · 103 + 7 · 102 + 8 · 10 + 9.
551 Example Find all whole numbers which begin with the digit 6 and decrease 25 times when this digit is deleted.
Solution: Let the number sought have n + 1 digits. Then this number can be written as 6 · 10n + y, where y is a number with n
digits (it may begin with one or several zeroes). The condition of the problem stipulates that
6 · 10n + y = 25 · y
whence
y=
6 · 10n
.
24
From this we gather that n ≥ 2 (otherwise, 6 · 10n would not be divisible by 24). For n ≥ 2, y = 25 · 10k−2 , that is, y has the form
250 · · · 0(n − 2 zeroes). We conclude that all the numbers sought have the form 625 0| .{z
. . 0} .
n−2 zeroes
552 Example (IMO 1968) Find all natural numbers x such that the product of their digits (in decimal notation) equals x2 −
10x − 22.
Solution: Let x have the form
x = a0 + a1 10 + a2 102 + · · · + an−1 10n−1 , ak ≤ 9, an−1 6= 0.
Let P(x) be the product of the digits of x, P(x) = x2 − 10x − 22. Now, P(x) = a0 a1 · · · an−1 ≤ 9n−1 an−1 < 10n−1 an−1 ≤ x (strict
inequality occurs when x has more than one digit). So x2 − 10x − 22 < x, and we deduce that x < 13, whence x has either one
digit or x = 10, 11, 13. If x had one digit, then a0 = x2 − 10x − 22, but this equation has no integral solutions. If x = 10, P(x) = 0,
but x2 − 10x − 22 6= 0. If x = 11, P(x) = 1, but x2 − 10x − 22 6= 1. If x = 12, P(x) = 2 and x2 − 10x − 22 = 2. Therefore, x = 12
is the only solution.
553 Example A whole number decreases an integral number of times when its last digit is deleted. Find all such numbers.
84
The Decimal Scale
85
Solution: Let 0 ≤ y ≤ 9, and 10x + y = mx, m and x natural numbers. This requires 10 + y/x = m, an integer. We must have
x|y. If y = 0, any natural number x will do, and we obtain the multiples of 10. If y = 1, x = 1, and we obtain 11. If y = 2, x = 1
or x = 2 and we obtain 12 and 22. Continuing in this fashion, the sought numbers are: the multiples of 10, 11, 12, 13, 14, 15,
16, 17, 18, 19,22, 24, 26, 28, 33, 36, 39, 44, 48, 55, 66, 77, 88, and 99.
554 Example Let A be a positive integer, and A′ be a number written with the aid of the same digits with are arranged in some
other order. Prove that if A + A′ = 1010 , then A is divisible by 10.
Solution: Clearly A and A′ must have ten digits. Let A = a10 a9 . . . a1 be the consecutive digits of A and A′ = a′10 a′9 . . . a′1 . Now,
A+A′ = 1010 if and only if there is a j, 0 ≤ j ≤ 9 for which a1 +a′1 = a2 +a′2 = · · · = a j +a′j = 0, a j+1 +a′j+1 = 10, a j+2 +a′j+2 =
a j+3 + a′j+3 = · · · = a10 + a′10 = 9. Notice that j = 0 implies that there are no sums of the form a j+k + a′j+k , k ≥ 2, and j = 9
implies that there are no sums of the form al + a′l , 1 ≤ l ≤ j. On adding all these sums, we gather
a1 + a′1 + a2 + a′2 + · · · + a10 + a′10 = 10 + 9(9 − j).
Since the a′s are a permutation of the as , we see that the sinistral side of the above equality is the even number 2(a1 + a2 + · · · +
a10 ). This implies that j must be odd. But this implies that a1 + a′1 = 0, which gives the result.
555 Example (AIME 1994) Given a positive integer n, let p(n) be the product of the non-zero digits of n. (If n has only one
digit, then p(n) is equal to that digit.) Let
S = p(1) + p(2) + · · · + p(999).
What is the largest prime factor of S?
Solution: Observe that non-zero digits are the ones that matter. So, for example, the numbers 180, 108, 118, 810, 800, and 811
have the same value p(n).
We obtain all the three digit numbers from 001 to 999 by expanding the product
(0 + 1 + 2 + · · · + 9)3 − 0,
where we subtracted a 0 in order to eliminate 000. Thus
(0 + 1 + 2 · · · + 9)3 − 0 = 001 + 002 + · · · + 999.
In order to obtain p(n) for a particular number, we just have to substitute the (possible) zeroes in the decimal representation, by
1’s, and so
p(1) + p(2) + · · · + p(n) = 111 + 112 + · · · + 999 = (1 + 1 + 2 + · · · + 9)3 − 1,
which equals 463 − 1. (In the last sum, 111 is repeated various times, once for 001, once for 011, once for 100, once for 101,
once for 110, etc.) As 463 − 1 = 33 · 5 · 7 · 103, the number required is 103.
556 Example (AIME 1992) Let S be the set of all rational numbers r, 0 < r < 1, that have a repeating decimal expansion of the
form
0.abcabcabc . . . = 0.abc,
where the digits a, b, c are not necessarily distinct. To write the elements of S as fractions in lowest terms, how many different
numerators are required?
abc
Solution: Observe that 0.abcabcabc . . . =
, and 999 = 33 · 37. If abc is neither divisible by 3 nor 37, the fraction is already
999
in lowest terms. By the Inclusion-Exclusion Principle, there are
999 − (999/3 + 999/37) + 999/3 · 37 = 648
such numbers. Also, fractions of the form s/37, where 3|s, 37 6 |s are in S. There are 12 fractions of this kind. (Observe that we
do not consider fractions of the form l/3t , 37|s, 3 6 |l, because fractions of this form are greater than 1, and thus not in S.)
The total number of distinct numerators in the set of reduced fractions is thus 640 + 12 = 660.
86
Chapter 8
557 Example (Putnam 1956) Prove that every positive integer has a multiple whose decimal representation involves all 10
digits.
Solution: Let n be an arbitrary positive integer with k digits. Let m = 123456780 · 10k+1 . Then all of the n consecutive integers
m + 1, m + 2, . . . m + n begin with 1234567890 and one of them is divisible by n.
558 Example (Putnam 1987) The sequence of digits
12345678910111213141516171819202122 . . .
is obtained by writing the positive integers in order. If the 10n digit of this sequence occurs in the part in which the m-digit
numbers are placed, define f (n) to be m. For example f (2) = 2, because the hundredth digit enters the sequence in the
placement of the two-digit integer 55. Find, with proof, f (1987).
Solution: There are 9 · 10 j−1 j-digit positive integers. The total number of digits in numbers with at most r digits is g(r) =
r
X
10r − 1
10r − 1
. As 0 <
< 10r , we get (r − 1)10r < g(r) < r10r . Thus g(1983) < 1983 · 101983 <
j · 9 · 10r−1 = r10r −
9
9
j=1
104 · 101983 = 101987 and g(1984) > 1983 · 101984 > 103 · 101984 . Therefore f (1987) = 1984.
Practice
559 Problem Prove that there is no whole number which decreases 35 times when its initial digit is deleted.
565 Problem Let t be a positive real number. Prove that there
is a positive integer n such that the decimal expansion of nt
contains a 7.
560 Problem A whole number is equal to the arithmetic mean
of all the numbers obtained from the given number with the aid 566 Problem (AIME 1988) Find the smallest positive integer
of all possible permutations of its digits. Find all whole num- whose cube ends in 888.
bers with that property.
567 Problem (AIME 1987) An ordered pair (m, n) of nonneg561 Problem (AIME 1989) Suppose that n is a positive inte-
ger and d is a single digit in base-ten. Find n if
n
= 0.d25d25d25d25 . . . .
810
ative integers is called simple if the addition m + n requires no
carrying. Find the number of simple ordered pairs of nonnegative integers that sum 1492.
568 Problem (AIME 1986) In the parlor game, the “magi-
562 Problem (AIME 1992) For how many pairs of consecu-
tive integers in
{1000, 1001, . . . , 2000}
is no carrying required when the two integers are added?
cian” asks one of the participants to think of a three-digit number abc, where a, b, c represent the digits of the number in the
order indicated. The magician asks his victim to form the numbers acb, bac, cab and cba, to add the number and to reveal
their sum N. If told the value of N, the magician can identity
abc. Play the magician and determine abc if N = 319.
563 Problem Let m be a seventeen-digit positive integer and
569 Problem The integer n is the smallest multiple of 15 such
let N be number obtained from m by writing the same digits
that every digit of n is either 0 or 8. Compute n/15.
in reversed order. Prove that at least one digit in the decimal
representation of the number M + N is even.
570 Problem (AIME 1988) For any positive integer k, let
f1 (k) denote the square of the sums of the digits of k. For
564 Problem Given that
n ≥ 2, let fn (k) = f1 ( fn−1 (k)). Find f1988 (11).
1
1
1
e = 2+ + + +··· ,
2! 3! 4!
571 Problem (IMO 1969) Determine all three-digit numbers
prove that e is irrational.
N that are divisible by 11 and such that N/11 equals the sum
Non-decimal Scales
87
of the squares of the digits of N.
574 Problem A Liouville number is a real number x such that
for every positive k there exist integers a and b ≥ 2, such that
572 Problem (IMO 1962) Find the smallest natural number
|x − a/b| < b−k .
having last digit is 6 and if this 6 is erased and put in front
of the other digits, the resulting number is four times as large
Prove or disprove that π is the sum of two Liouville numbers.
as the original number.
573 Problem
1. Show that Champernowne’s number
575 Problem Given that
χ = 0.123456789101112131415161718192021 . . .
1/49 = 0.020408163265306122448979591836734693877551,
is irrational.
2. Let r ∈ Q and let ε > 0 be given. Prove that there exists
a positive integer n such that
find the last thousand digits of
|10n χ − r| < ε .
1 + 50 + 502 + · · · + 50999 .
8.2 Non-decimal Scales
The fact that most people have ten fingers has fixed our scale of notation to the decimal. Given any positive integer r > 1, we
can, however, express any number in base r.
576 Example Express the decimal number 5213 in base-seven.
Solution: Observe that 5213 < 75 . We thus want to find 0 ≤ a0 , . . . , a4 ≤ 6, a4 6= 0, such that
5213 = a4 74 + a3 73 + a2 72 + a1 7 + a0 .
Now, divide by 74 to obtain
2 + proper fraction = a4 + proper fraction.
Since a4 is an integer, it must be the case that a4 = 2. Thus 5213 − 2 · 74 = 411 = a3 73 + a2 72 + a1 7 + a0 . Dividing 411 by 73
we obtain
1 + proper fraction = a3 + proper fraction.
Thus a3 = 1. Continuing in this way we deduce that 5213 = 211257 .
577 Example Express the decimal number 13/16 in base-six.
Solution: Write
13 a1 a2 a3
=
+ + +....
16
6 62 63
Multiply by 6 to obtain
4 + proper fraction = a1 + proper fraction.
a2 a3
Thus a1 = 4. Hence 13/16 − 4/6 = 7/48 = 2 + 3 + . . .. Multiply by 62 to obtain
6
6
5 + proper fraction = a2 + proper fraction.
We gather that a2 = 5. Continuing in this fashion, we deduce that 13/16 = .45136 .
578 Example Prove that 4.41 is a perfect square in any scale of notation.
Solution: If 4.41 is in scale r, then

4.41 = 4 +
1
4 1
+ 2 = 2+
r r
r
‹2
.
88
Chapter 8
579 Example Let TxU denote the greatest integer less than or equal to x. Does the equation
TxU + T2xU + T4xU + T8xU + T16xU + T32xU = 12345
have a solution?
Solution: We show that there is no such x. Recall that TxU satisfies the inequalities x − 1 < TxU ≤ x. Thus
x − 1 + 2x − 1 + 4x − 1 + · · · + 32x − 1
< TxU + T2xU + T4xU + T8xU
+T16xU + T32xU
≤ x + 2x + 4x + · · · + 32x.
From this we see that 63x − 6 < 12345 ≤ 63x. Hence 195 < x < 196.
Write then x in base-two:
a1 a2 a3
x = 195 + + 2 + 3 + . . . ,
2 2
2
with ak = 0 or 1. Then
T2xU
= 2 · 195 + a1 ,
T4xU
= 4 · 195 + 2a1 + a2 ,
T8xU
= 8 · 195 + 4a1 + 2a2 + a3 ,
T16xU = 16 · 195 + 8a1 + 4a2 + 2a3 + a4 ,
T32xU = 32 · 195 + 16a1 + 8a2 + 4a3 + 2a4 + a5 .
Adding we find that TxU + T2xU + T4xU + T8xU + T16xU + T32xU = 63 · 195 + 31a1 + 15a2 + 7a3 + 3a4 + a5 , i.e. 31a1 + 15a2 +
7a3 + 3a4 + a5 = 60. This cannot be because 31a1 + 15a2 + 7a3 + 3a4 + a5 ≤ 31 + 15 + 7 + 3 + 1 = 57 < 60.
580 Example (AHSME 1993) Given 0 ≤ x0 < 1, let
§
xn =
2xn−1
if 2xn−1 < 1
2xn−1 − 1 if 2xn−1 ≥ 1
for all integers n > 0. For how many x0 is it true that x0 = x5 ?
Solution: Write x0 in base-two,
x0 =
∞
X
k=1
an
an = 0 or 1.
2n
The algorithm given just moves the binary point one unit to the right. For x0 to equal x5 we need 0.a1 a2 a3 a4 a5 a6 a7 . . . =
0.a6 a7 a8 a9 a10 a11 a12 . . .. This will happen if and only if x0 has a repeating expansion with a1 a2 a3 a4 a5 as the repeating block .
There are 25 = 32 such blocks. But if a1 = a2 = · · · = a5 = 1, then x0 = 1, which is outside [0, 1). The total number of values
for which x0 = x5 is thus 32 − 1 = 31.
581 Example (AIME 1986) The increasing sequence
1, 3, 4, 9, 10, 12, 13, . . .
consists of all those positive integers which are powers of 3 or sums distinct powers of 3. Find the hundredth term of the
sequence.
Solution: If the terms of the sequence are written in base-3, they comprise the positive integers which do not contain the digit
2. Thus, the terms of the sequence in ascending order are thus
1, 10, 11, 100, 101, 110, 111, . . . .
In the binary scale, these numbers are, of course, 1, 2, 3, . . . . To obtain the 100-th term of the sequence we just write 100 in
binary 100 = 11001002 and translate this into ternary: 11001003 = 36 + 35 + 32 = 981.
Practice
A theorem of Kummer
89
where xi′ = 2xi if xi 6= x j and x′j = 2x j − 1. If the new triple is
not balanced, one performs the balancing act on it. Does continuation of this process always lead to a balanced triple after
a finite number of performances of the balancing act?
582 Problem (Putnam, 1987) For each positive integer n, let
α (n) be the number of zeroes in the base-three representation
of n. For which positive real numbers x does the series
∞
X
n=1
xα (n)
n3
587 Problem Let C denote the class of positive integers
which, when written in base-three, do not require the digit 2.
Show that no three integers in C are in arithmetic progression.
converge?
588 Problem Let B(n) be the number of 1’s in the base-two
expansion of n. For example, B(6) = B(1102 ) = 2, B(15) =
B(11112 ) = 4.
583 Problem Prove that for x ∈ R, x ≥ 0, one has
∞
X
n=1
(−1)T2
2n
n xU
= 1 − 2(x − TxU).
1. (P UTNAM 1981) Is
∞
X
exp
584 Problem (Putnam, 1981) Let E(n) denote the largest k
such that 5k is an integral divisor of 11 22 33 · · · nn . Calculate
n=1
!
B(n)
n2 + n
a rational number?
E(n)
.
n→∞ n2
lim
2. (P UTNAM 1984) Express
m −1
2X
585 Problem (AHSME, 1982) The base-eight representation
of a perfect square is ab3c with a 6= 0. Find the value of c.
(−1)B(n) nm
n=0
in the form (−1)m a f (m) (g(m))! where a is an integer
ordered
triple
of
and f , g are polynomials.
(x1 , x2 , x3 ) of positive irrational numbers with x1 + x2 + x3 = 1
is called balanced if xn < 1/2 for all 1 ≤ n ≤ 3. If a triple is not
balanced, say x j > 1/2, one performs the following “balancing 589 Problem What is the largest integer that I should be peract”:
mitted to choose so that you may determine my number in
twenty “yes” or “no” questions?
B(x1 , x2 , x3 ) = (x1′ , x2′ , x3′ ),
586 Problem (Putnam, 1977) An
8.3 A theorem of Kummer
We first establish the following theorem.
590 Theorem (Legendre) Let p be a prime and let n = a0 pk + a1 pk−1 + · · · + ak−1 p + ak be the base-p expansion of n. The
exact power m of a prime p dividing n! is given by
m=
n − (a0 + a1 + · · · + ak )
.
p−1
Proof: By De Polignac’s Formula
m=
∞
X
T
k=1
n
U.
pk
90
Chapter 8
Now, Tn/pU = a0 pk−1 + a1 pk−2 + · · · ak−2 p + ak−1 , Tn/p2 U = a0 pk−2 + a1 pk−3 + · · · + ak−2 , . . . , Tn/pk U = a0 .
Thus
∞
X
Tn/pk U =
k=1
=
=
=
a0 (1 + p + p2 + · · · + pk−1 ) + a1 (1 + p + p2 + · · · + pk−2 )+
· · · + ak−1 (1 + p) + ak
pk − 1
pk−1 − 1
p2 − 1
p−1
a0
+ a1
+ · · · + ak−1
+ ak
p−1
p−1
p−1
p−1
a0 pk + a1 pk−1 + · · · + ak − (a0 + a1 + · · · + ak )
p−1
n − (a0 + a1 + · · · + ak )
,
p−1
as wanted.❑
591 Theorem (Kummer’s Theorem) The exact power of a prime p dividing the binomial coefficient
a+b
a
is equal to the
number of “carry-overs” when performing the addition of a, b written in base p.
Proof:
k
X
j=0
Let a = a0 + a1 p + · · · + ak pk , b = b0 + b1 p + · · · + bk pk , 0 ≤ a j , b j ≤ p − 1, and ak + bk > 0. Let Sa =
a j , Sb =
k
X
j=0
b j . Let c j , 0 ≤ c j ≤ p − 1, and ε j = 0 or 1, be defined as follows:
a0 + b0 = ε0 p + c0 ,
ε0 + a1 + b1 = ε1 p + c1 ,
ε1 + a2 + b2 = ε2 p + c2 ,
..
.
εk−1 + ak + bk = εk p + ck .
Multiplying all these equalities successively by 1, p, p2 , . . . and adding them:
a + b + ε0 p + ε1 p2 + . . . + εk−1 pk
=
ε0 p + ε1 p2 + . . . + εk−1 pk + εk pk+1
.
+c0 + c1 p + · · · + ck pk
We deduce that a + b = c0 + c1 p + · · · + ck pk + εk pk+1 . By adding all the equalities above, we obtain similarly:
Sa + Sb + (ε0 + ε1 + · · · + εk−1 ) = (ε0 + ε1 + · · · + εk )p + Sa+b − εk .
Upon using Legendre’s result from above,
(p − 1)m = (a + b) − Sa+b − a + Sa − b + Sb = (p − 1)(ε0 + ε1 + · · · + εk ),
which gives the result.❑
Chapter
9
Miscellaneous Problems
592 Example Prove that
X
1
p
p
p prime
diverges.
Solution: Let Fx denote the family consisting of the integer 1 and the positive integers n all whose prime factors are less than
or equal to x. By the Unique Factorisation Theorem

Y
‹
1+
p≤x
1
1
+
+··· =
p p2
p prime
Now,
X
n∈Fx
X
n∈Fx
1
.
n
(9.1)
1 X1
>
.
n n≤x n
As the harmonic series diverges, the product on the sinistral side of 2.3.3 diverges as x → ∞. But
Y

‹
1+
p≤x
1
1
+ 2 +··· =
p p
p prime
X
p≤x
1
+ O(1).
p
p prime
This finishes the proof.
593 Example Prove that for each positive integer k there exist infinitely many even positive integers which can be written in
more than k ways as the sum of two odd primes.
Solution: Let ak denote the number of ways in which 2k can be written as the sum of two odd primes. Assume that ak ≤ C ∀ k
for some positive constant C. Then
†
2
X
xp
p>2
p prime
This yields
X
p>2
p prime
=
∞
X
k=2
x p−1 ≤
ak x2k ≤ C
x4
.
1 − x2
√
x
.
C√
1 − x2
91
92
Chapter 9
Integrating term by term,
X
p>2
p prime
1 √
≤ C
p
Z 1
0
√
x
√
dx = C.
1 − x2
But the leftmost series is divergent, and we obtain a contradiction.
594 Example (IMO 1976) Determine, with proof, the largest number which is the product of positive integers whose sum is
1976.
Solution: Suppose that
a1 + a2 + · · · + an = 1976;
n
Y
ak . We shall replace some of the ak so that the product is enlarged, but the sum remains the same. By
we want to maximise
k=1
the arithmetic mean-geometric mean inequality
!1/n
n
Y
ak
k=1
≤
a1 + a2 + · · · + an
,
n
with equality if and only if a1 = a2 = · · · = an . Thus we want to make the ak as equal as possible.
If we have an ak ≥ 4, we replace it by two numbers 2, ak − 2. Then the sum is not affected, but 2(ak − 2) ≥ ak , since we are
assuming ak ≥ 4. Therefore, in order to maximise the product, we must take ak = 2 or ak = 3. We must take as many 2’s and
3’s as possible.
Now, 2 + 2 + 2 = 3 + 3, but 23 < 32 , thus we should take no more than two 2’s. Since 1976 = 3 · 658 + 2, the largest possible
product is 2 · 3658 .
595 Example (USAMO 1983) Consider an open interval of length 1/n on the real line, where n is a positive integer. Prove that
the number of irreducible fractions a/b, 1 ≤ b ≤ n, contained in the given interval is at most (n + 1)/2.
sk
}, k = 1, 2, . . . , r, with denominators 1 ≤ tk ≤ n/2 and
tk
those uk /vk , k = 1, 2, . . . , s with denominators n/2 < vk ≤ n, where all these fractions are in reduced form. Now, for every tk
there are integers ck such that n/2 ≤ ck tk ≤ n. Define us+k = ck sk , vs+k = ck tk , yk+r = uk+r /vk+r . No two of the yl , 1 ≤ l ≤ r + s
are equal, for otherwise y j = yk would yield
Solution: Divide the rational numbers in (x, x + 1/n) into two sets: {
|uk /vk − ui /vi | ≥ 1/vi ≥ 1/n,
which contradicts that the open interval is of length 1/n. Hence the number of distinct rationals is r +s ≤ n−Tn/2U ≤ (n+1)/2.
Aliter: Suppose to the contrary that we have at least T(n + 1)/2U + 1 = a fractions. Let sk ,tk , 1 ≤ k ≤ a be the set of
numerators and denominators. The set of denominators is a subset of
{1, 2, . . . , 2(a − 1)}.
By the Pigeonhole Principle, ti |tk for some i, k, say tk = mti . But then
|sk /tk − si /ti | = |msi − sk |/tk ≥ 1/n,
contradicting the hypothesis that the open interval is of length 1/n.
596 Example Let
Qr,s =
Show that Qr,ps ≡ Qr,s mod p, where p is a prime
(rs)!
.
r!s!
Practice
93
Solution: As
r
Y
Qr,s =
j=1
and
Qr,ps =
r
Y
j=1
js − 1
s−1
j ps − 1
,
ps − 1
it follows from
(1 + x) jps−1 ≡ (1 + x p ) js−1 (1 + x) p−1
that
j ps − 1
≡
ps − 1
mod p
js − 1
s−1
mod p,
whence the result.
Practice
597 Problem Find a four-digit number which is a perfect
square such that its first two digits are equal to each other and
its last two digits are equal to each other.
604 Problem Determine two-parameter solutions for the “almost” Fermat Diophantine equations
xn−1 + yn−1 = zn ,
598 Problem Find all integral solutions of the equation
x
X
xn+1 + yn+1 = zn ,
k! = y2 .
xn+1 + yn−1 = zn .
k=1
599 Problem Find all integral solutions of the equation
x
X
k! = yz .
605 Problem (AIME 1984) What is the largest even integer
which cannot be written as the sum of two odd composite numbers?
k=1
600 Problem (USAMO 1985) Determine whether there are
any positive integral solutions to the simultaneous equations
2
= y3 ,
x12 + x22 + · · · + x1985
3
x13 + x23 + · · · + x1985
= z2
606 Problem Prove that are infinitely many nonnegative inte-
gers n which cannot be written as n = x2 + y3 + z6 for nonnegative integers x, y, z.
607 Problem Find the integral solutions of
with distinct integers x1 , x2 , . . . , x1985 .
x2 + x = y4 + y3 + y2 + y.
601 Problem Show that the Diophantine equation
1
1
1
1
1
+ +...+
+ +
a1 a2
an−1 an a1 a2 · · · an
608 Problem Show that there are infinitely many integers x, y
such that
3x2 − 7y2 = −1.
has at least one solution for every n ∈ N.
602 Problem (AIME 1987) Find the largest possible value of
k for which 311 is expressible as the sum of k consecutive positive integers.
603 Problem (AIME 1987) Let M be the smallest positive
integer whose cube is of the form n + r, where n ∈ N, 0 < r <
1/1000. Find n.
609 Problem Prove that
1.
a3 +b3 +c3 −3abc = (a+b+c)(a2 +b2 +c2 −ab−bc−ca).
2. Find integers a, b, c such that 1987 = a3 +b3 +c3 −3abc.
94
Chapter 9
3. Find polynomials P, Q, R in x, y, z such that
619 Problem (Wostenholme’s Theorem) Let p > 3 be a
prime. If
P3 + Q3 + R3 − 3PQR = (x3 + y3 + z3 − 3xyz)2
4. Can you find integers a, b, c with 19872 = a3 + b3 + c3 −
3abc?
610 Problem Find all integers n such that n4 + n + 7 is a per-
a
1 1
1
= 1+ + +···+
,
b
2 3
p−1
then p2 |a.
620 Problem Prove that the number of odd binomial coefficients in any row of Pascal’s Triangle is a power of 2.
fect square.
621 Problem Prove that the coefficients of a binomial expan611 Problem Prove that 19911991 is not the sum of two per-
sion are odd if and only if n is of the form 2k − 1.
fect squares.
622 Problem Let the numbers ci be defined by the power se612 Problem Find infinitely many integers x > 1, y > 1, z > 1
such that
x!y! = z!.
613 Problem Find all positive integers with
mn − nm = 1.
614 Problem Find all integers with
x4 − 2y2 = 1.
615 Problem Prove that for every positive integer k there
exists a sequence of k consecutive positive integers none of
which can be represented as the sum of two squares.
ries identity
(1 + x + x2 + · · · + x p−1 )/(1 − x) p−1 := 1 + c1 x + c2 x2 + · · · .
Show that ci ≡ 0 mod p for all i ≥ 1.
623 Problem Let p be a prime. Show that
p−1
≡ (−1)k
k
mod p
for all 0 ≤ k ≤ p − 1.
624 Problem (Putnam 1977) Let p be a prime and let a ≥
b > 0 be integers. Prove that
pa
a
≡
pb
b
mod p.
616 Problem (IMO 1977) In a finite sequence of real num-
bers, the sum of any seven successive terms is negative, and
the sum of any eleven successive terms is positive. Determine
the maximum number of terms in the sequence.
625 Problem Demonstrate that for a prime p and k ∈ N,
pk
a
≡0
mod p,
617 Problem Determine an infinite series of terms such that for 0 < a < pk .
each term of the series is a perfect square and the sum of the
series at any point is also a perfect square.
626 Problem Let p be a prime and let k, a ∈ N, 0 ≤ a ≤ pk −1.
Demonstrate that
618 Problem Prove that any positive rational integer can be
pk − 1
expressed as a finite sum of distinct terms of the harmonic se≡ (−1)a mod p.
a
ries, 1, 1/2, 1/3, . . ..